#7527249

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=i,d^2=i,e^2=f,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, 7434445);