#7330559

order	512
nilpotency class	3
isoclinism	isoclinic class of #7324500
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	#53051	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1528 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fg,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=ehi,d^c=dhi,e^c=ei,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, 7237755);