#7584760

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	#23975	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1569 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)(19,25)(20,26)(21,33)(22,24)(23,27)(28,38)(29,40)(30,36)(31,42)(32,37)(34,39)(35,41)(43,48)(44,47)(45,46);(2,6)(7,8)(9,11,12,16)(10,13,14,15)(18,24)(20,31)(21,23)(25,36)(26,41)(28,43)(29,34)(32,35)(37,42)(38,47)(44,45)(46,48);(1,3,5,4)(2,7,6,8)(10,14)(11,16)(17,19)(18,25)(20,28)(21,34)(22,30)(23,29)(24,36)(26,38)(27,40)(31,43)(32,45)(33,39)(35,44)(37,46)(41,47)(42,48);(10,14)(13,15)(17,20,22,32)(18,26,24,37)(19,28,30,45)(21,31,23,35)(25,38,36,46)(27,41,33,42)(29,44,34,43)(39,48,40,47);(1,4,5,3)(2,7,6,8)(10,14)(13,15)(17,21,22,23)(18,27,24,33)(19,29,30,34)(20,31,32,35)(25,39,36,40)(26,41,37,42)(28,44,45,43)(38,48,46,47)

Polycyclic presentation:

2^9|a^2=fi,b^2=g,c^2=fgi,d^2=i,e^2=fg,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, 7491956);