#10158110

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

4994 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)(6,8)(17,18,22,23)(19,25,26,20)(21,29,31,24)(27,28,32,30);(1,2,5,7)(3,8,4,6)(9,10)(11,16)(12,14)(13,15)(18,23)(20,25)(24,29)(28,30);(1,3,5,4)(2,6,7,8)(9,11,13,12)(10,14,15,16)(18,23)(19,26)(24,29)(27,32);(1,2,5,7)(3,8,4,6)(9,10)(11,14)(12,16)(13,15)(17,19,22,26)(18,20,23,25)(21,27,31,32)(24,28,29,30);(1,2)(3,6)(4,8)(5,7)(9,12,13,11)(10,14,15,16)(17,20)(18,19)(21,30)(22,25)(23,26)(24,32)(27,29)(28,31);(1,4,5,3)(2,6,7,8)(9,10)(11,14)(12,16)(13,15)(17,21,22,31)(18,24,23,29)(19,27,26,32)(20,28,25,30)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=i,d^2=h,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10065306);