#7410060

order	512
nilpotency class	3
isoclinism	isoclinic class of #7300980
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×#547	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1589 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)(9,10,13,12)(11,14,15,16)(17,18)(19,24)(20,30)(21,26)(22,37)(23,27)(25,28)(29,39)(31,41)(32,46)(33,38)(34,47)(35,43)(36,48)(40,44)(42,45);(1,2)(3,8)(4,6)(5,7)(9,11)(10,14)(12,16)(13,15)(17,19,23,22)(18,24,27,37)(20,31,33,28)(21,29,35,36)(25,30,41,38)(26,39,43,48)(32,45,34,44)(40,46,42,47);(10,12)(11,15)(17,20,23,33)(18,25,27,41)(19,28,22,31)(21,34,35,32)(24,38,37,30)(26,42,43,40)(29,45,36,44)(39,46,48,47);(1,3,5,4)(2,6,7,8)(9,12,13,10)(11,14,15,16)(17,21,23,35)(18,26,27,43)(19,29,22,36)(20,32,33,34)(24,39,37,48)(25,40,41,42)(28,44,31,45)(30,46,38,47);(1,4,5,3)(2,6,7,8)(11,15)(14,16)(20,33)(21,35)(25,41)(26,43)(28,31)(29,36)(30,38)(39,48)

Polycyclic presentation:

2^9|a^2=fhi,b^2=i,c^2=i,d^2=fh,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfhi,e^b=ehi,d^c=dh,e^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7317256);