#7609408

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518944
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	#2388	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1751 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)(19,24)(20,32)(21,26)(22,27)(23,28)(25,34)(29,42)(30,40)(31,41)(33,39)(35,45)(36,46)(37,47)(38,43)(44,48);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19,22,30)(18,24,27,40)(20,33,34,29)(21,35,36,48)(23,38,37,31)(25,42,32,39)(26,45,46,44)(28,43,47,41);(10,14)(11,15)(17,20,22,34)(18,25,27,32)(19,29,30,33)(21,31,36,38)(23,35,37,48)(24,39,40,42)(26,43,46,41)(28,44,47,45);(1,3,4,7)(2,6,5,8)(10,14)(13,16)(17,21,23,29,22,36,37,33)(18,26,28,42,27,46,47,39)(19,20,31,35,30,34,38,48)(24,32,41,45,40,25,43,44);(2,5)(6,8)(20,34)(21,36)(25,32)(26,46)(29,33)(35,48)(39,42)(44,45)

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=fh,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,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7516604);