#7798072

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

4377 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=gh,d^2=gi,e^2=gi,f^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,e^b=ei,d^c=dh,f^c=fi,e^d=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, 7705268);