#6507177

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500456
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#25131	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1665 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6414373);