#8533574

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

4404 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=gi,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=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, 8440770);