#7508644

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500669
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	#2540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1442 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)(3,4)(5,7)(6,8)(9,10)(11,15)(12,14)(13,16)(17,18,23,27)(19,24,30,39)(20,25,33,44)(21,34,22,26)(28,37,46,47)(29,40,35,38)(31,41,45,48)(32,43,36,42);(1,3)(2,6)(4,7)(5,8)(17,19,23,30)(18,24,27,39)(20,31,33,45)(21,29,22,35)(25,41,44,48)(26,38,34,40)(28,36,46,32)(37,42,47,43);(9,11,12,16)(10,13,14,15)(17,20)(18,25)(19,28)(21,32)(22,36)(23,33)(24,37)(26,42)(27,44)(29,45)(30,46)(31,35)(34,43)(38,41)(39,47)(40,48);(2,7)(3,8)(10,14)(13,15)(17,21,23,22)(18,26,27,34)(19,29,30,35)(20,32,33,36)(24,38,39,40)(25,42,44,43)(28,45,46,31)(37,41,47,48);(1,4,5,6)(2,3,7,8)(18,27)(20,33)(24,39)(26,34)(28,46)(31,45)(32,36)(38,40)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=g,d^2=i,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dfgi,e^a=ei,c^b=ch,d^b=df,h^b=hi,e^c=ei,h^c=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7415840);