#7584786

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 ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1526 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=g,c^2=fg,d^2=i,e^2=fgi,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, 7491982);