#7584766

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514020
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	C₂×#555	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1548 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)(19,24,31,38)(20,25,33,40)(21,26,22,27)(29,36,45,48)(30,37,34,41)(32,39,35,42)(43,46,44,47);(2,6)(7,8)(9,11,12,16)(10,13,14,15)(20,32)(21,22)(25,39)(26,27)(29,43)(30,34)(33,35)(36,46)(37,41)(40,42)(44,45)(47,48);(1,3,5,4)(2,7,6,8)(10,14)(11,16)(17,19)(18,24)(20,29)(21,34)(22,30)(23,31)(25,36)(26,41)(27,37)(28,38)(32,43)(33,45)(35,44)(39,46)(40,48)(42,47);(10,14)(13,15)(17,20,23,33)(18,25,28,40)(19,29,31,45)(21,32,22,35)(24,36,38,48)(26,39,27,42)(30,44,34,43)(37,47,41,46);(1,4,5,3)(2,7,6,8)(10,14)(13,15)(17,21,23,22)(18,26,28,27)(19,30,31,34)(20,32,33,35)(24,37,38,41)(25,39,40,42)(29,44,45,43)(36,47,48,46)

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=dg,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

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7491962);