#7527250

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
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 ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1418 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7434446);