#496501

order	512
nilpotency class	4
isoclinism	isoclinic class of #481393
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#60682	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

826 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

(2,5)(3,7)(9,10,13,18,28,27,39,25,15,19,29,17,14,12,24,26)(11,16,22,32,38,34,40,35,21,30,37,31,23,20,33,36);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,20)(13,22)(14,23)(15,21)(17,31)(18,32)(19,30)(24,33)(25,35)(26,36)(27,34)(28,38)(29,37)(39,40);(2,5)(6,8)(9,12)(10,14)(11,20)(13,18)(15,27)(16,23)(17,29)(19,28)(21,34)(22,32)(24,25)(26,39)(30,38)(31,37)(33,35)(36,40);(1,3,4,7)(2,6,5,8)(11,21)(16,30)(20,34)(22,37)(23,38)(31,32)(33,40)(35,36)

Polycyclic presentation:

2^9|a^2=g,d^2=e,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=ce,d^b=di,f^c=fh,g^c=ghi,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265458, #72265476, #72265498, …
maximal normal subgroup of	#72265458, #72265476, #72265498, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403697);