#8912422

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

5027 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=g,e^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=egi

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, 8819618);