#7709921

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

4474 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)(9,10,15,12)(11,16,14,18)(13,20,22,17)(19,24,21,23)(25,26,31,28)(27,32,35,36)(29,39,38,33)(30,40,37,34);(1,2)(3,7)(4,8)(5,6)(9,11,15,14)(10,16,12,18)(13,19,22,21)(17,23,20,24)(25,27)(26,32)(28,36)(29,37)(30,38)(31,35)(33,40)(34,39);(1,3,5,4)(2,7,6,8)(10,12)(16,18)(17,20)(23,24)(26,28)(27,35)(29,38)(34,40);(9,12,15,10)(11,18,14,16)(13,20,22,17)(19,24,21,23)(25,28,31,26)(27,36,35,32)(29,39,38,33)(30,40,37,34);(1,3,5,4)(2,7,6,8)(9,13,15,22)(10,17,12,20)(11,19,14,21)(16,23,18,24)(25,29,31,38)(26,33,28,39)(27,37,35,30)(32,40,36,34);(1,4,5,3)(2,8,6,7)(9,14,15,11)(10,18,12,16)(13,21,22,19)(17,24,20,23)(25,30,31,37)(26,34,28,40)(27,38,35,29)(32,39,36,33)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=g,d^2=h,e^2=gh,f^2=gh,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, 7617117);