#6964500

order	512
nilpotency class	3
isoclinism	isoclinic class of #6884969
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1334 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=fi,c^2=f,d^2=i,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=dfh,g^c=gi,e^d=ei

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6871696);