#7737406

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

4505 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)(25,26)(27,32)(28,34)(29,30)(31,33)(35,39)(36,38)(37,40);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,19)(13,14)(16,20)(17,21)(18,24)(22,23)(25,27,31,36)(26,32,33,38)(28,29,37,39)(30,40,35,34);(1,3,4,7)(2,6,5,8)(10,14)(11,17)(16,19)(18,23)(26,33)(27,36)(29,39)(34,40);(9,10)(11,15)(12,16)(13,14)(17,21)(18,22)(19,20)(23,24)(25,28,31,37)(26,34,33,40)(27,29,36,39)(30,38,35,32);(9,11,13,17)(10,15,14,21)(12,18,20,23)(16,22,19,24)(25,29)(26,35)(27,37)(28,36)(30,33)(31,39)(32,34)(38,40);(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)(25,30,31,35)(26,29,33,39)(27,34,36,40)(28,38,37,32)

Polycyclic presentation:

2^9|b^2=h,c^2=g,d^2=h,e^2=hi,f^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,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, 7644602);