#10324602

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

4804 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)(3,7)(4,6)(5,8)(9,10)(11,12)(13,14)(15,16)(17,18,22,26)(19,23,27,32)(20,24,30,29)(21,25,31,28);(1,3,4,8)(2,5,6,7)(10,14)(11,15)(19,27)(21,31)(23,32)(25,28);(1,2,4,6)(3,5,8,7)(9,11,13,15)(10,12,14,16)(17,19)(18,23)(20,25)(21,29)(22,27)(24,31)(26,32)(28,30);(1,3)(2,5)(4,8)(6,7)(9,12)(10,15)(11,14)(13,16)(17,19)(18,23)(20,25)(21,29)(22,27)(24,31)(26,32)(28,30);(1,2)(3,5)(4,6)(7,8)(17,20)(18,24)(19,28)(21,23)(22,30)(25,27)(26,29)(31,32);(1,3)(2,5)(4,8)(6,7)(17,21)(18,25)(19,29)(20,23)(22,31)(24,27)(26,28)(30,32)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^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, 10231798);