#8189405

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

5003 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=h,d^2=hi,e^2=g,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, 8096601);