#7683195

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

5177 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7590391);