#7283745

order	512
nilpotency class	3
isoclinism	isoclinic class of #7234485
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1678 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(2,5)(3,7)(17,18,23,24,22,27,19,28)(20,31,29,41,33,38,37,25)(21,34,35,42,36,26,30,39)(32,45,43,40,47,44,46,48);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,23)(18,24,27,28)(20,29,33,37)(21,35,36,30)(25,38,41,31)(26,42,34,39)(32,46,47,43)(40,44,48,45);(10,13)(11,15)(17,20,22,33)(18,25,27,41)(19,29,23,37)(21,32,36,47)(24,38,28,31)(26,40,34,48)(30,43,35,46)(39,45,42,44);(2,5)(6,8)(9,11,12,15)(10,14,13,16)(17,21)(18,26)(19,30)(20,32)(22,36)(23,35)(24,39)(25,40)(27,34)(28,42)(29,43)(31,44)(33,47)(37,46)(38,45)(41,48);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(21,36)(26,34)(30,35)(32,47)(39,42)(40,48)(43,46)(44,45)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,d^2=h,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=efi,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7190941);