#499683

order	512
nilpotency class	4
isoclinism	isoclinic class of #13373
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#486
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#23815
prime radical	#3150	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

505 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 505 conjugacy classes of subgroups — too many to draw. Only its 144 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2)(3,10)(4,22,25,42)(5,12)(6,13)(7,30)(8,15)(9,16)(11,26,40,27)(14,28)(17,47,49,63)(18,35)(19,36)(20,53)(21,37)(23,57,58,55)(24,56,59,45)(29,43)(31,61)(32,41,60,39)(33,46)(34,50,62,52)(38,54,64,48)(44,51);(1,3)(2,10)(4,23,25,58)(5,18)(6,19)(7,31)(8,33)(9,21)(11,38,40,64)(12,35)(13,36)(14,44)(15,46)(16,37)(17,24,49,59)(20,29)(22,55,42,57)(26,48,27,54)(28,51)(30,61)(32,52,60,50)(34,41,62,39)(43,53)(45,63,56,47);(1,4)(2,11)(3,17)(5,24)(6,27)(7,32)(8,26)(9,25)(10,34)(12,39)(13,42)(14,45)(15,22)(16,40)(18,48)(19,52)(20,54)(21,49)(23,51)(28,60)(29,59)(30,56)(31,58)(33,50)(35,57)(36,63)(37,62)(38,61)(41,43)(44,64)(46,47)(53,55);(1,5,6,28,9,29,8,7)(2,12,13,14,16,43,15,30)(3,18,19,51,21,20,33,31)(4,24,27,60,25,59,26,32)(10,35,36,44,37,53,46,61)(11,39,42,45,40,41,22,56)(17,48,52,23,49,54,50,58)(34,57,63,64,62,55,47,38)

Polycyclic presentation:

2^9|a^2=e,b^2=e,d^2=ef,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,g^a=gi,c^b=cg,g^b=gh,h^b=hi,e^c=ei,f^c=fi,g^c=ghi,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283823, #72283870, #72283942, …
maximal normal subgroup of	#72283823, #72283870, #72283942, …
maximal quotient of	C₂×#499683

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406879);