#8912608

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

4638 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)(3,8,4,6)(17,18)(19,20)(21,29)(22,23)(24,25)(26,30)(27,31)(28,32);(3,4)(6,8)(9,10)(11,16)(12,14)(13,15)(18,23)(19,25)(26,29)(27,32);(1,3,5,4)(2,6,7,8)(9,11)(10,12)(13,14)(15,16)(17,19,22,25)(18,24,23,20)(21,27,30,32)(26,31,29,28);(3,4)(6,8)(11,14)(12,16)(17,20,22,24)(18,25,23,19)(21,28,30,31)(26,32,29,27);(1,4,5,3)(2,6,7,8)(9,12,13,16)(10,14,15,11)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,32)(29,30);(1,2)(3,8)(4,6)(5,7)(9,10)(11,16)(12,14)(13,15)(17,21)(18,26)(19,27)(20,28)(22,30)(23,29)(24,31)(25,32)

Polycyclic presentation:

2^9|a^2=gh,c^2=h,d^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, 8819804);