#8726303

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

5354 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)(3,4,5,7)(11,15)(12,16);(1,3,6,5)(2,7,8,4)(9,10,14,13)(11,16,15,12)(17,18,22,23)(19,24,26,20)(21,25,31,32)(27,30,29,28);(1,3,6,5)(2,7,8,4)(9,11)(10,12)(13,16)(14,15)(18,23)(19,26)(25,32)(27,29);(2,8)(3,5)(10,13)(11,15)(17,19)(18,24)(20,23)(21,29)(22,26)(25,28)(27,31)(30,32);(1,4)(2,3)(5,8)(6,7)(9,12)(10,11)(13,15)(14,16)(17,20,22,24)(18,19,23,26)(21,30,31,28)(25,29,32,27);(1,5,6,3)(2,7,8,4)(9,13,14,10)(11,16,15,12)(17,21,22,31)(18,25,23,32)(19,27,26,29)(20,28,24,30)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=g,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8633499);