#7620502

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518964
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

1232 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)(17,18,22,27)(19,29,31,24)(20,33,36,25)(21,38,32,26)(23,28,39,45)(30,41,34,42)(35,43,40,46)(37,47,48,44);(2,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,31)(18,24,27,29)(20,34,36,30)(21,39,32,23)(25,42,33,41)(26,45,38,28)(35,48,40,37)(43,47,46,44);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,20,22,36)(18,25,27,33)(19,30,31,34)(21,40,32,35)(23,37,39,48)(24,41,29,42)(26,46,38,43)(28,44,45,47);(1,4,5,3)(2,8,6,7)(10,13)(14,16)(17,21,23,19,22,32,39,31)(18,26,28,24,27,38,45,29)(20,35,37,34,36,40,48,30)(25,43,44,42,33,46,47,41);(2,6)(7,8)(9,11,12,15)(10,14,13,16)(18,27)(20,36)(24,29)(26,38)(28,45)(30,34)(35,40)(37,48)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=f,d^2=fgh,e^2=gi,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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7527698);