#7632550

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5251 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=gi,d^2=hi,e^2=i,f^2=g,b^a=bg,c^a=ch,c^b=ci,f^b=fi,e^d=ei

Conjugacy classes & character table

122 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7539746);