#10324294

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

4708 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=h,d^2=gh,e^2=h,f^2=h,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, 10231490);