#7609816

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

1747 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)(11,13)(12,14)(15,16)(17,18)(19,24)(20,33)(21,28)(22,27)(23,31)(25,36)(26,38)(29,39)(30,40)(32,42)(34,43)(35,44)(37,45)(41,46)(47,48);(1,3,5,4)(2,6,7,8)(9,11)(10,13)(12,15)(14,16)(17,19)(18,23)(20,34)(21,30)(22,31)(24,27)(25,43)(26,40)(28,42)(29,37)(32,38)(33,48)(35,46)(36,47)(39,44)(41,45);(10,14)(11,15)(17,19)(18,24)(20,29)(21,32)(22,31)(23,27)(25,41)(26,40)(28,42)(30,38)(33,39)(34,37)(35,47)(36,46)(43,45)(44,48);(1,4,5,3)(2,6,7,8)(10,14)(13,16)(17,20,21,37,22,36,38,35)(18,25,28,44,27,33,26,45)(19,29,32,34,31,46,30,47)(23,39,40,43,24,41,42,48);(2,7)(6,8)(17,21,22,38)(18,26,27,28)(19,30,31,32)(20,35,36,37)(23,40,24,42)(25,44,33,45)(29,34,46,47)(39,48,41,43)

Polycyclic presentation:

2^9|b^2=fi,d^2=fhi,e^2=i,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, 7517012);