#7715451

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

4333 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7622647);