#10328302

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

4582 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=gh,d^2=gh,e^2=h,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, 10235498);