#7534606

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	C₂×#547	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1533 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)(11,15)(12,14)(13,16)(17,18)(19,25)(20,35)(21,26)(22,27)(23,28)(24,29)(30,44)(31,40)(32,41)(33,42)(34,43)(36,39)(37,46)(38,45)(47,48);(2,6)(7,8)(11,16)(13,15)(17,19,23,33)(18,25,28,42)(20,36,21,30)(22,34,24,32)(26,44,35,39)(27,43,29,41)(31,37,47,38)(40,46,48,45);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,13,14,15)(17,20,23,21)(18,26,28,35)(19,30,33,36)(22,38,24,37)(25,39,42,44)(27,46,29,45)(31,34,47,32)(40,41,48,43);(10,14)(13,15)(17,21,23,20)(18,26,28,35)(19,31,33,47)(22,38,24,37)(25,40,42,48)(27,45,29,46)(30,34,36,32)(39,41,44,43);(1,4,5,3)(2,7,6,8)(17,22,23,24)(18,27,28,29)(19,32,33,34)(20,37,21,38)(25,41,42,43)(26,45,35,46)(30,47,36,31)(39,40,44,48)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fg,d^2=i,e^2=fi,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, 7441802);