#7608603

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
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×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1647 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)(11,13)(12,14)(15,16)(17,18,23,22)(19,24,29,36)(20,25,31,32)(21,33,35,26)(27,37,41,46)(28,42,39,34)(30,43,45,38)(40,48,47,44);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19,23,29)(18,24,22,36)(20,27,31,41)(21,34,35,42)(25,37,32,46)(26,39,33,28)(30,44,45,48)(38,47,43,40);(1,3,4,7)(2,6,5,8)(10,14)(11,15)(17,20,23,31)(18,25,22,32)(19,27,29,41)(21,30,35,45)(24,37,36,46)(26,38,33,43)(28,40,39,47)(34,44,42,48);(10,14)(13,16)(17,21,18,26,23,35,22,33)(19,28,36,42,29,39,24,34)(20,30,25,38,31,45,32,43)(27,40,46,48,41,47,37,44);(2,5)(6,8)(20,31)(25,32)(27,41)(30,45)(37,46)(38,43)(40,47)(44,48)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fi,d^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,h^b=hi,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, 7515799);