#8197104

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

4993 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)(11,12)(13,14)(15,16)(17,18,23,22)(19,26,24,20)(21,29,30,25)(27,28,31,32);(1,3,5,4)(2,6,7,8)(10,14)(11,15)(17,19,23,24)(18,20,22,26)(21,27,30,31)(25,28,29,32);(2,7)(6,8)(9,11,13,15)(10,12,14,16)(17,20,23,26)(18,24,22,19)(21,28,30,32)(25,31,29,27);(1,3,5,4)(2,6,7,8)(9,12)(10,15)(11,14)(13,16)(17,19)(18,20)(21,27)(22,26)(23,24)(25,28)(29,32)(30,31);(1,4,5,3)(2,8,7,6)(17,21,23,30)(18,25,22,29)(19,27,24,31)(20,28,26,32);(1,2)(3,6)(4,8)(5,7)(17,22,23,18)(19,26,24,20)(21,29,30,25)(27,32,31,28)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=i,d^2=gi,e^2=h,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8104300);