#7584720

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	C₂×#553	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1551 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)(17,18,22,26)(19,23,27,34)(20,24,31,38)(21,25,32,39)(28,35,41,45)(29,36,42,46)(30,37,33,40)(43,47,44,48);(2,6)(7,8)(9,11,12,16)(10,13,14,15)(19,27)(20,30)(21,32)(23,34)(24,37)(25,39)(28,43)(31,33)(35,47)(38,40)(41,44)(45,48);(1,3,5,4)(2,7,6,8)(10,14)(11,16)(17,19,22,27)(18,23,26,34)(20,28,31,41)(21,29,32,42)(24,35,38,45)(25,36,39,46)(30,44,33,43)(37,48,40,47);(10,14)(13,15)(17,20,22,31)(18,24,26,38)(19,28,27,41)(21,33,32,30)(23,35,34,45)(25,40,39,37)(29,43,42,44)(36,47,46,48);(1,4,5,3)(2,7,6,8)(10,14)(13,15)(19,27)(23,34)(28,41)(29,42)(35,45)(36,46)(43,44)(47,48)

Polycyclic presentation:

2^9|a^2=fi,b^2=g,c^2=fgi,d^2=i,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cgi,d^b=dh,h^b=hi,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, 7491916);