#6389704

order	512
nilpotency class	3
isoclinism	isoclinic class of #62283
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×C₄×D₄
prime radical	#60685	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#433 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1618 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=fg,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ci,d^b=df,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6296900);