#10322938

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

4337 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,d^2=g,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, 10230134);