#6395647

order	512
nilpotency class	3
isoclinism	isoclinic class of #61609
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	#3455	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#433 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1571 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,d^2=f,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=dfi,g^c=gi,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, 6302843);