#7590150

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1581 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)(17,18,22,26)(19,23,28,36)(20,30,32,24)(21,25,33,39)(27,41,43,35)(29,37,44,47)(31,40,34,38)(42,48,45,46);(2,6)(7,8)(9,11,12,16)(10,13,14,15)(20,31)(21,33)(24,38)(25,39)(27,42)(29,44)(30,40)(32,34)(35,46)(37,47)(41,48)(43,45);(1,3,5,4)(2,7,6,8)(10,14)(11,16)(17,19)(18,23)(20,27)(21,29)(22,28)(24,35)(25,37)(26,36)(30,41)(31,42)(32,43)(33,44)(34,45)(38,46)(39,47)(40,48);(1,3,5,4)(2,7,6,8)(10,14)(13,15)(17,20,22,32)(18,24,26,30)(19,27,28,43)(21,34,33,31)(23,35,36,41)(25,40,39,38)(29,45,44,42)(37,48,47,46);(1,4,5,3)(2,7,6,8)(10,14)(13,15)(19,28)(23,36)(27,43)(29,44)(35,41)(37,47)(42,45)(46,48)

Polycyclic presentation:

2^9|a^2=fi,b^2=g,c^2=fg,d^2=fgi,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,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, 7497346);