#7656540

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

4816 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=gh,c^2=g,d^2=gh,e^2=gi,f^2=gi,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, 7563736);