#8070072

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

4719 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:

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

Polycyclic presentation:

2^9|a^2=h,d^2=h,f^2=g,b^a=bg,c^a=ch,e^a=ei,c^b=ci,e^c=ei,f^c=fh,e^d=ehi,f^d=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7977268);