#7710815

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

4477 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7618011);