#9070563

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

4937 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8977759);