#8093148

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

4540 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)(10,14)(15,20)(16,21)(22,24)(25,26,31,29)(27,34,33,30)(28,36,37,32)(35,38,39,40);(2,5)(6,8)(11,17)(12,19)(15,20)(16,21)(26,29)(27,33)(28,37)(38,40);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,27)(26,30)(28,35)(29,34)(31,33)(32,38)(36,40)(37,39);(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24)(25,28,31,37)(26,32,29,36)(27,35,33,39)(30,38,34,40);(1,2)(3,6)(4,5)(7,8)(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)(25,29,31,26)(27,34,33,30)(28,36,37,32)(35,40,39,38);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,30,31,34)(26,33,29,27)(28,38,37,40)(32,39,36,35)

Polycyclic presentation:

2^9|a^2=g,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^b=eghi

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, 8000344);