#7576322

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, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1228 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fi,d^2=fi,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, 7483518);