#10332167

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

4958 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)(9,10,13,11)(12,15,16,14)(17,18,22,25)(19,20,23,28)(21,24,30,32)(26,27,29,31);(2,6)(4,7)(10,11)(14,15)(17,19,22,23)(18,20,25,28)(21,29,30,26)(24,31,32,27);(1,3,5,8)(2,7,6,4)(9,11,13,10)(12,15,16,14)(17,19)(18,20)(21,26)(22,23)(24,27)(25,28)(29,30)(31,32);(1,4)(2,3)(5,7)(6,8)(9,12)(10,14)(11,15)(13,16)(17,19)(18,20)(21,26)(22,23)(24,27)(25,28)(29,30)(31,32);(1,3)(2,4)(5,8)(6,7)(17,20)(18,23)(19,25)(21,27)(22,28)(24,29)(26,32)(30,31);(1,4)(2,3)(5,7)(6,8)(17,21)(18,24)(19,26)(20,27)(22,30)(23,29)(25,32)(28,31)

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,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, 10239363);