#9028428

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

4636 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=ghi,d^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,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, 8935624);