#6389705

order	512
nilpotency class	3
isoclinism	isoclinic class of #62307
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	#60682	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#433 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1623 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)(17,18,23,28,24,29,22,40)(19,25,32,43,33,44,31,47)(20,34,30,42,35,48,36,26)(21,27,39,46,37,45,38,41);(1,3)(2,7)(4,11)(5,13)(6,12)(8,15)(9,10)(14,16)(17,19,23,32,24,33,22,31)(18,25,28,43,29,44,40,47)(20,21,30,39,35,37,36,38)(26,41,34,27,42,46,48,45);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,15,10,16)(17,20,22,36,24,35,23,30)(18,26,28,34,29,42,40,48)(19,21,31,38,33,37,32,39)(25,41,43,27,44,46,47,45);(3,12)(7,10)(11,13)(15,16)(17,21)(18,27)(19,30)(20,31)(22,38)(23,39)(24,37)(25,42)(26,44)(28,46)(29,45)(32,35)(33,36)(34,47)(40,41)(43,48);(1,5,6,4)(2,8,9,14)(3,13,12,11)(7,15,10,16)(20,35)(21,37)(26,42)(27,45)(30,36)(34,48)(38,39)(41,46)

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,d^b=df,e^c=ei,e^d=ei

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6296901);