#10324467

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

4367 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

(9,10)(11,14)(12,15)(13,16)(17,19)(18,21)(20,22)(23,24)(25,26,31,28)(27,34,30,32)(29,33,38,36)(35,40,37,39);(1,2)(3,7)(4,5)(6,8)(10,11)(13,21)(15,17)(20,24)(25,26,31,28)(27,32,30,34)(29,33,38,36)(35,39,37,40);(2,5)(6,7)(12,19)(15,17)(20,24)(22,23)(25,27,31,30)(26,32,28,34)(29,35,38,37)(33,39,36,40);(2,5)(6,7)(9,11,14,10)(12,15,19,17)(13,18,21,16)(20,22,24,23)(25,28,31,26)(27,34,30,32)(29,33,38,36)(35,39,37,40);(1,2)(3,6)(4,5)(7,8)(9,12,14,19)(10,15,11,17)(13,20,21,24)(16,22,18,23)(25,29,31,38)(26,33,28,36)(27,35,30,37)(32,39,34,40);(1,3)(2,6)(4,8)(5,7)(9,13,14,21)(10,16,11,18)(12,20,19,24)(15,22,17,23)(25,30,31,27)(26,34,28,32)(29,37,38,35)(33,40,36,39)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,d^2=gh,e^2=gh,f^2=gh,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, 10231663);