#8188075

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

4913 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=gh,d^2=ghi,e^2=gh,f^2=h,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, 8095271);