#7336643

order	512
nilpotency class	3
isoclinism	isoclinic class of #7324820
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	#2535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1501 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=gi,d^2=fi,e^2=fi,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^d=ei

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7243839);