#7575014

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1304 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)(3,8)(4,6)(5,7)(9,10)(11,14)(12,13)(15,16)(17,18,23,29)(19,25,33,42)(20,27,21,26)(22,37,24,28)(30,44,34,39)(31,48,47,40)(32,43,38,41)(35,45,36,46);(1,3,5,4)(2,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,25)(20,34)(21,30)(22,32)(23,33)(24,38)(26,44)(27,39)(28,41)(29,42)(31,36)(35,47)(37,43)(40,45)(46,48);(1,4,5,3)(2,6,7,8)(10,13)(14,16)(17,20)(18,26)(19,30)(21,23)(22,36)(24,35)(25,39)(27,29)(28,46)(31,38)(32,47)(33,34)(37,45)(40,41)(42,44)(43,48);(10,13)(14,16)(17,21,23,20)(18,27,29,26)(19,31,33,47)(22,36,24,35)(25,40,42,48)(28,46,37,45)(30,32,34,38)(39,43,44,41);(2,7)(6,8)(9,11,12,15)(10,14,13,16)(17,22,23,24)(18,28,29,37)(19,32,33,38)(20,35,21,36)(25,41,42,43)(26,45,27,46)(30,31,34,47)(39,48,44,40)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=f,d^2=i,e^2=g,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, 7482210);