#6435060

order	512
nilpotency class	3
isoclinism	isoclinic class of #6432881
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3385 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,d^2=h,f^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,c^b=ch,f^b=fi,g^b=gi,f^c=fi,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318324
maximal normal subgroup of	#72318324

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6342256);