#7544219

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×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1504 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)(3,6)(4,5)(7,8)(9,10,12,14)(11,15,16,13)(17,18)(19,27)(20,31)(21,25)(22,26)(23,28)(24,34)(29,37)(30,43)(32,42)(33,40)(35,41)(36,39)(38,44)(45,47)(46,48);(2,5)(6,8)(9,11,12,16)(10,13,14,15)(19,28)(20,32)(22,35)(23,27)(24,39)(26,41)(29,45)(31,42)(33,46)(34,36)(37,47)(40,48);(1,3,4,7)(2,6,5,8)(10,14)(11,16)(17,19,21,28)(18,23,25,27)(20,33,34,29)(22,30,35,44)(24,40,31,37)(26,38,41,43)(32,45,36,46)(39,47,42,48);(10,14)(13,15)(17,20,21,34)(18,24,25,31)(19,29,28,33)(22,36,35,32)(23,37,27,40)(26,42,41,39)(30,45,44,46)(38,47,43,48);(2,5)(6,8)(10,14)(13,15)(18,25)(19,28)(24,31)(26,41)(29,33)(30,44)(39,42)(45,46)

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=fg,d^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, 7451415);