#8115225

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

4397 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)(12,20)(16,21)(18,24)(22,23)(25,26,30,33)(27,31,35,29)(28,36,38,32)(34,37,40,39);(3,7)(6,8)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,27,30,35)(26,29,33,31)(28,34,38,40)(32,37,36,39);(1,2,5,4)(3,6,7,8)(10,14)(15,17)(16,21)(22,23)(25,27)(26,31)(28,34)(29,33)(30,35)(32,39)(36,37)(38,40);(1,3,5,7)(2,6,4,8)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,28,30,38)(26,32,33,36)(27,34,35,40)(29,37,31,39);(1,4,5,2)(3,8,7,6)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23)(25,29,30,31)(26,27,33,35)(28,37,38,39)(32,34,36,40);(9,12,13,20)(10,16,14,21)(11,18,19,24)(15,22,17,23)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=gh,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,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, 8022421);