#7234139

order	512
nilpotency class	3
isoclinism	isoclinic class of #7207830
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 ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1499 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=fghi,d^2=fh,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,d^c=di,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7141335);