#8196988

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

5005 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,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, 8104184);