#7621207

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1257 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)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,18,23,29)(19,30,33,25)(20,26,36,43)(21,37,32,27)(22,38,24,28)(31,45,46,40)(34,48,47,41)(35,44,39,42);(2,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,23,33)(18,25,29,30)(20,31,36,46)(21,22,32,24)(26,40,43,45)(27,38,37,28)(34,39,47,35)(41,42,48,44);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,20)(18,26)(19,31)(21,34)(22,39)(23,36)(24,35)(25,40)(27,41)(28,44)(29,43)(30,45)(32,47)(33,46)(37,48)(38,42);(10,13)(14,16)(17,21,24,19,23,32,22,33)(18,27,28,25,29,37,38,30)(20,34,35,31,36,47,39,46)(26,41,44,40,43,48,42,45);(1,4,5,3)(2,7,6,8)(9,11,12,15)(10,14,13,16)(17,22,23,24)(18,28,29,38)(19,32,33,21)(20,35,36,39)(25,27,30,37)(26,42,43,44)(31,34,46,47)(40,48,45,41)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fi,d^2=gh,e^2=fg,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, 7528403);