#6948326

order	512
nilpotency class	3
isoclinism	isoclinic class of #6884841
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×#466	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1302 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6855522);