#7656804

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

4771 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)(3,8)(4,6)(5,7)(9,10,13,16)(11,14,12,15)(17,18,24,23)(19,28,26,21)(20,25,31,29)(22,27,32,30);(9,10)(11,14)(12,15)(13,16)(17,19)(18,21)(20,30)(22,29)(23,28)(24,26)(25,32)(27,31);(1,3,5,6)(2,4,7,8)(17,20,24,31)(18,25,23,29)(19,27,26,30)(21,22,28,32);(9,11,13,12)(10,14,16,15)(17,21,24,28)(18,26,23,19)(20,22,31,32)(25,30,29,27);(1,2)(3,4)(5,7)(6,8)(9,12,13,11)(10,15,16,14)(17,22)(18,27)(19,29)(20,28)(21,31)(23,30)(24,32)(25,26);(1,4)(2,6)(3,7)(5,8)(17,23,24,18)(19,28,26,21)(20,29,31,25)(22,27,32,30)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7564000);