#7732813

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

4590 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)(25,26,31,32)(27,34,28,30)(29,37,38,33)(35,39,36,40);(1,2)(3,6)(4,5)(7,8)(25,27,31,28)(26,30,32,34)(29,35,38,36)(33,39,37,40);(10,14)(15,17)(16,21)(22,23)(26,32)(27,28)(29,38)(39,40);(1,3,4,7)(2,6,5,8)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,28,31,27)(26,30,32,34)(29,36,38,35)(33,39,37,40);(9,11,13,19)(10,15,14,17)(12,18,20,24)(16,22,21,23)(25,29)(26,33)(27,35)(28,36)(30,39)(31,38)(32,37)(34,40);(9,12,13,20)(10,16,14,21)(11,18,19,24)(15,22,17,23)(25,30)(26,27)(28,32)(29,39)(31,34)(33,35)(36,37)(38,40)

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=g,e^2=hi,f^2=hi,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7640009);