#7795709

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

4598 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)(9,10,15,14)(11,19,20,16)(12,17,22,23)(13,24,21,18)(25,26,31,28)(27,35,37,32)(29,38,40,33)(30,34,36,39);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,21)(13,22)(14,19)(15,20)(17,24)(18,23)(25,27,31,37)(26,32,28,35)(29,36,40,30)(33,39,38,34);(1,3,4,7)(2,6,5,8)(10,14)(11,20)(12,22)(18,24)(26,28)(32,35)(33,38)(34,39);(9,12,15,22)(10,17,14,23)(11,21,20,13)(16,24,19,18)(25,28,31,26)(27,35,37,32)(29,38,40,33)(30,39,36,34);(9,13,15,21)(10,18,14,24)(11,12,20,22)(16,17,19,23)(25,29)(26,33)(27,36)(28,38)(30,37)(31,40)(32,39)(34,35);(9,14,15,10)(11,19,20,16)(12,23,22,17)(13,24,21,18)(25,30,31,36)(26,34,28,39)(27,29,37,40)(32,33,35,38)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=gh,d^2=h,e^2=i,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, 7702905);