#7589855

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

1562 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)(17,18,21,25)(19,27,30,23)(20,24,32,40)(22,26,34,42)(28,45,47,37)(29,46,48,38)(31,43,35,39)(33,41,36,44);(2,6)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19,21,30)(18,23,25,27)(20,28,32,47)(22,35,34,31)(24,37,40,45)(26,43,42,39)(29,36,48,33)(38,44,46,41);(1,3,5,4)(2,7,6,8)(10,14)(11,16)(17,20,21,32)(18,24,25,40)(19,28,30,47)(22,33,34,36)(23,37,27,45)(26,41,42,44)(29,31,48,35)(38,39,46,43);(1,3,5,4)(2,7,6,8)(10,14)(13,15)(17,20)(18,24)(19,29)(21,32)(22,36)(23,38)(25,40)(26,44)(27,46)(28,35)(30,48)(31,47)(33,34)(37,43)(39,45)(41,42);(1,4,5,3)(2,7,6,8)(10,14)(13,15)(20,32)(24,40)(28,47)(29,48)(33,36)(37,45)(38,46)(41,44)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fg,d^2=fgi,e^2=fgi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,e^d=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, 7497051);