#7714993

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

4579 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)(13,21)(16,20)(18,24)(22,23)(25,26)(27,33)(28,36)(29,30)(31,35)(32,37)(34,38)(39,40);(1,2)(3,7)(4,8)(5,6)(9,10,14,12)(11,17,19,15)(13,16,21,20)(18,23,24,22)(26,30)(31,33)(32,36)(38,39);(10,12)(15,17)(16,20)(22,23)(25,26,29,30)(27,33,35,31)(28,32,37,36)(34,39,40,38);(9,11,14,19)(10,15,12,17)(13,18,21,24)(16,22,20,23)(25,26)(27,31)(28,32)(29,30)(33,35)(34,38)(36,37)(39,40);(1,3,5,4)(2,7,6,8)(9,12,14,10)(11,17,19,15)(13,20,21,16)(18,23,24,22)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,39)(37,40);(1,4,5,3)(2,8,6,7)(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,28,29,37)(26,32,30,36)(27,34,35,40)(31,38,33,39)

Polycyclic presentation:

2^9|b^2=hi,c^2=h,d^2=hi,e^2=gi,f^2=g,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, 7622189);