#8096307

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

4389 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8003503);