#6404305

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	C₂×#135	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#446 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1535 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)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18,22,27,23,28,37,26)(19,24,31,41,32,42,47,40)(20,33,21,25,34,43,36,44)(29,45,35,38,46,48,30,39);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,19)(18,24)(20,29)(21,35)(22,31)(23,32)(25,38)(26,40)(27,41)(28,42)(30,36)(33,45)(34,46)(37,47)(39,44)(43,48);(17,20,22,21,23,34,37,36)(18,25,26,33,28,44,27,43)(19,29,31,35,32,46,47,30)(24,38,40,45,42,39,41,48);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15);(2,8)(6,9)(7,16)(14,15)(17,21,22,34,23,36,37,20)(18,25,27,43,28,44,26,33)(19,30,31,29,32,35,47,46)(24,39,41,45,42,38,40,48)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=g,d^2=f,e^2=h,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=df,e^b=ei,e^c=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, 6311501);