#7589858

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×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1550 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)(19,24)(20,31)(21,35)(22,27)(23,26)(25,34)(28,42)(29,41)(30,39)(32,38)(33,43)(36,44)(37,40)(45,47)(46,48);(2,6)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19,22,30)(18,24,27,39)(20,32,34,28)(21,29,23,37)(25,42,31,38)(26,40,35,41)(33,46,36,45)(43,48,44,47);(1,3,5,4)(2,7,6,8)(10,14)(11,16)(17,20)(18,25)(19,28)(21,36)(22,34)(23,33)(24,38)(26,44)(27,31)(29,46)(30,32)(35,43)(37,45)(39,42)(40,48)(41,47);(1,3,5,4)(2,7,6,8)(10,14)(13,15)(17,21)(18,26)(20,33)(22,23)(24,39)(25,43)(27,35)(28,32)(29,37)(31,44)(34,36)(47,48);(1,4,5,3)(2,7,6,8)(10,14)(13,15)(17,21,22,23)(18,26,27,35)(19,29,30,37)(20,33,34,36)(24,40,39,41)(25,43,31,44)(28,45,32,46)(38,47,42,48)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fgi,d^2=fgi,e^2=fg,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, 7497054);