#6378449

order	512
nilpotency class	2
isoclinism	isoclinic class of #40484
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	#392	C₂×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1360 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)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,21,25,22,26,32,38)(19,23,29,35,27,33,40,44)(20,24,31,37,30,36,42,46)(28,34,41,45,39,43,47,48);(2,5)(3,7)(9,11,12,15)(10,13,14,16)(20,30)(24,36)(28,39)(31,42)(34,43)(37,46)(41,47)(45,48);(1,3,4,7)(2,6,5,8)(10,14)(13,16)(17,19)(18,23)(20,28)(21,29)(22,27)(24,34)(25,35)(26,33)(30,39)(31,41)(32,40)(36,43)(37,45)(38,44)(42,47)(46,48);(2,5)(6,8)(19,27)(23,33)(28,39)(29,40)(34,43)(35,44)(41,47)(45,48);(17,20,21,31,22,30,32,42)(18,24,25,37,26,36,38,46)(19,28,29,41,27,39,40,47)(23,34,35,45,33,43,44,48)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=f,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=cf,e^b=ei,d^c=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6285645);