#7532228

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
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	#45332	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1710 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)(11,15)(12,14)(13,16)(17,18,23,28)(19,24,32,40)(20,25,34,42)(21,26,22,27)(29,37,46,48)(30,38,45,47)(31,39,36,44)(33,41,35,43);(2,5)(6,8)(11,16)(13,15)(17,19,23,32)(18,24,28,40)(20,29,34,46)(21,31,22,36)(25,37,42,48)(26,39,27,44)(30,35,45,33)(38,43,47,41);(9,11,12,16)(10,13,14,15)(17,20)(18,25)(19,29)(21,35)(22,33)(23,34)(24,37)(26,43)(27,41)(28,42)(30,36)(31,45)(32,46)(38,44)(39,47)(40,48);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,20)(18,25)(19,30)(21,33)(22,35)(23,34)(24,38)(26,41)(27,43)(28,42)(29,36)(31,46)(32,45)(37,44)(39,48)(40,47);(2,5)(6,8)(17,21,23,22)(18,26,28,27)(19,31,32,36)(20,33,34,35)(24,39,40,44)(25,41,42,43)(29,30,46,45)(37,38,48,47)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=g,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei,h^d=hi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7439424);