#7540110

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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1519 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)(17,18,23,27)(19,24,30,40)(20,25,34,43)(21,35,22,26)(28,37,45,48)(29,39,36,38)(31,41,46,47)(32,44,33,42);(2,5)(6,8)(9,11,12,16)(10,13,14,15)(17,19)(18,24)(20,31)(21,29)(22,36)(23,30)(25,41)(26,38)(27,40)(28,33)(32,45)(34,46)(35,39)(37,42)(43,47)(44,48);(1,3,4,7)(2,6,5,8)(10,14)(11,16)(17,19)(18,24)(20,28)(21,36)(22,29)(23,30)(25,37)(26,39)(27,40)(31,33)(32,46)(34,45)(35,38)(41,42)(43,48)(44,47);(10,14)(13,15)(17,20,23,34)(18,25,27,43)(19,28,30,45)(21,33,22,32)(24,37,40,48)(26,44,35,42)(29,46,36,31)(38,41,39,47);(2,5)(6,8)(10,14)(13,15)(17,21,23,22)(18,26,27,35)(19,29,30,36)(20,32,34,33)(24,38,40,39)(25,42,43,44)(28,31,45,46)(37,47,48,41)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=fg,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,h^b=hi,e^c=ei,e^d=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, 7447306);