#7917105

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
prime radical	C₂×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4161 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)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,26,31,33)(27,37,38,32)(28,30,36,34)(29,40,39,35);(1,2)(3,6)(4,5)(7,8)(25,27)(26,32)(28,39)(29,36)(30,40)(31,38)(33,37)(34,35);(1,3,4,7)(2,6,5,8)(10,14)(15,17)(16,21)(22,23)(26,33)(27,38)(28,36)(35,40);(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,28,31,36)(26,34,33,30)(27,39,38,29)(32,35,37,40);(9,11,13,19)(10,15,14,17)(12,18,20,24)(16,22,21,23)(25,29,31,39)(26,35,33,40)(27,36,38,28)(30,32,34,37);(9,12,13,20)(10,16,14,21)(11,18,19,24)(15,22,17,23)(25,30,31,34)(26,36,33,28)(27,40,38,35)(29,37,39,32)

Polycyclic presentation:

2^9|a^2=i,c^2=gi,d^2=i,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=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, 7824301);