#499478

order	512
nilpotency class	4
isoclinism	isoclinic class of #499261
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3184
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#23682
prime radical	#13340	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

573 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 573 conjugacy classes of subgroups — too many to draw. Only its 134 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283942, C₂×#499478
maximal normal subgroup of	#72283942, C₂×#499478
maximal quotient of	C₂×#499478

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406674);