#8195191

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

4769 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=g,d^2=gh,e^2=h,f^2=gi,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, 8102387);