#8295597

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

4566 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8202793);