#10038492

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

4488 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=gh,d^2=gi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi

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, 9945688);