#10320495

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

4992 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10227691);