#7569777

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	#35771	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1716 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)(3,7)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(17,18,23,27)(19,24,30,41)(20,25,34,46)(21,35,22,26)(28,38,32,43)(29,40,36,39)(31,42,47,48)(33,45,37,44);(2,6)(7,8)(9,11,12,16)(10,13,14,15)(17,19)(18,24)(20,31)(21,29)(22,36)(23,30)(25,42)(26,39)(27,41)(28,33)(32,37)(34,47)(35,40)(38,45)(43,44)(46,48);(1,3,5,4)(2,7,6,8)(10,14)(11,16)(17,19)(18,24)(20,32)(21,36)(22,29)(23,30)(25,43)(26,40)(27,41)(28,34)(31,37)(33,47)(35,39)(38,46)(42,44)(45,48);(1,4,5,3)(2,8,6,7)(10,14)(13,15)(17,20)(18,25)(19,28)(21,33)(22,37)(23,34)(24,38)(26,44)(27,46)(29,47)(30,32)(31,36)(35,45)(39,42)(40,48)(41,43);(2,6)(7,8)(10,14)(13,15)(17,21,23,22)(18,26,27,35)(19,29,30,36)(20,33,34,37)(24,39,41,40)(25,44,46,45)(28,47,32,31)(38,42,43,48)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=fg,d^2=fg,e^2=i,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,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, 7476973);