#496252

order	512
nilpotency class	4
isoclinism	isoclinic class of #496249
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#21750
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×#41	C₂⁴
prime radical	#3499	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #3499 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

836 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 836 conjugacy classes of subgroups — too many to draw. Only its 184 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)(3,14,19,41)(4,20)(5,25,27,12)(6,11)(7,13)(8,17,34,28)(9,22,38,59)(10,26,40,61)(15,46,49,62)(16,39)(18,32)(21,55,51,64)(23,53)(24,54)(29,36)(30,43)(31,56)(33,52)(35,48)(37,58,42,60)(44,50)(45,47)(57,63);(1,3)(2,8)(4,21)(5,16)(6,28)(7,19)(9,35)(10,39)(11,41)(12,44)(13,34)(14,43)(15,23)(17,30)(18,27)(20,46)(22,56)(24,51)(25,33)(26,52)(29,49)(31,37)(32,40)(36,55)(38,45)(42,63)(47,58)(48,60)(50,61)(53,64)(54,62)(57,59);(1,4,7,24)(2,9,13,38)(3,15,19,49)(5,22,27,59)(6,29,30,23)(8,31,34,63)(10,36,40,53)(11,42,43,37)(12,20,25,54)(14,45,41,35)(16,47,18,48)(17,51,28,21)(26,60,61,58)(32,62,39,46)(33,55,44,64)(50,57,52,56);(1,5,7,27)(2,10)(3,16,19,18)(4,22,24,59)(6,26,30,61)(8,32)(9,36)(11,25)(12,43)(13,40)(14,33)(15,47,49,48)(17,50,28,52)(20,37)(21,56,51,57)(23,58,29,60)(31,62)(34,39)(35,64)(38,53)(41,44)(42,54)(45,55)(46,63)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403448);