#7065227

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996799
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#61175	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1762 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6972423);