#7584446

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514020
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

1254 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)(3,6,7,8)(9,10)(11,12)(13,15)(14,16)(17,18,23,29)(19,30,34,25)(20,27,21,26)(22,38,24,28)(31,40,35,44)(32,41,48,47)(33,42,39,43)(36,45,37,46);(2,5)(6,8)(9,11,14,15)(10,12,16,13)(17,19)(18,25)(20,35)(21,31)(22,33)(23,34)(24,39)(26,44)(27,40)(28,42)(29,30)(32,37)(36,48)(38,43)(41,45)(46,47);(1,3,4,7)(2,6,5,8)(9,12,14,13)(10,15,16,11)(17,20)(18,26)(19,31)(21,23)(22,37)(24,36)(25,40)(27,29)(28,46)(30,44)(32,39)(33,48)(34,35)(38,45)(41,42)(43,47);(10,16)(11,15)(17,21,23,20)(18,27,29,26)(19,32,34,48)(22,37,24,36)(25,41,30,47)(28,46,38,45)(31,33,35,39)(40,43,44,42);(2,5)(6,8)(9,13,14,12)(10,15,16,11)(17,22,23,24)(18,28,29,38)(19,33,34,39)(20,36,21,37)(25,42,30,43)(26,45,27,46)(31,32,35,48)(40,47,44,41)

Polycyclic presentation:

2^9|a^2=f,b^2=gi,c^2=fg,d^2=i,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,h^d=hi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7491642);