#8182912

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

4527 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,d^2=h,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8090108);