#7543431

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1327 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)(3,7)(4,8)(5,6)(9,10)(11,14)(12,16)(13,15)(17,18)(19,29)(20,25)(21,37)(22,28)(23,27)(24,33)(26,36)(30,44)(31,40)(32,41)(34,45)(35,43)(38,42)(39,48)(46,47);(2,6)(7,8)(9,11,14,10)(12,16,13,15)(17,19)(18,24)(20,30)(21,34)(22,33)(23,32)(25,39)(26,44)(27,45)(28,29)(31,35)(36,48)(37,41)(38,47)(40,42)(43,46);(1,3,5,4)(2,7,6,8)(10,11)(15,16)(17,20)(18,25)(19,30)(21,38)(22,36)(23,35)(24,39)(26,28)(27,43)(29,44)(31,32)(33,48)(34,47)(37,42)(40,41)(45,46);(1,3,5,4)(2,7,6,8)(9,12,14,13)(10,15,11,16)(17,20)(18,26)(19,31)(21,35)(22,36)(23,38)(24,40)(25,28)(27,43)(29,46)(30,32)(33,47)(34,48)(37,42)(39,41)(44,45);(1,4,5,3)(2,7,6,8)(9,13,14,12)(10,16,11,15)(17,21,22,23)(18,27,28,37)(19,32,33,34)(20,35,36,38)(24,41,29,45)(25,42,26,43)(30,47,48,31)(39,46,44,40)

Polycyclic presentation:

2^9|b^2=g,c^2=fgi,d^2=fi,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,e^b=ei,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, 7450627);