#7569651

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	#36009	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1779 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=fg,d^2=fgi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,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, 7476847);