#8757080

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

4408 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=hi,d^2=g,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8664276);