#9041011

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

5343 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8948207);