#8108314

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

4536 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:

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

Polycyclic presentation:

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

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, 8015510);