#6376604

order	512
nilpotency class	3
isoclinism	isoclinic class of #60756
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×#405 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1653 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)(3,9)(4,8)(5,7)(6,10)(11,15)(12,14)(13,16)(17,18,23,28,24,29,22,39)(19,25,31,41,32,26,20,33)(21,27,38,44,36,43,37,48)(30,40,35,47,45,42,34,46);(1,3)(2,6)(4,11)(5,10)(7,9)(8,13)(12,16)(14,15)(17,19,23,31,24,32,22,20)(18,25,28,41,29,26,39,33)(21,30,38,35,36,45,37,34)(27,40,44,47,43,42,48,46);(3,10)(6,9)(11,16)(13,15)(17,20,22,32,24,31,23,19)(18,26,28,33,29,25,39,41)(21,35,37,30,36,34,38,45)(27,40,44,47,43,42,48,46);(2,7)(6,9)(8,14)(13,15)(17,21)(18,27)(19,30)(20,34)(22,37)(23,38)(24,36)(25,40)(26,42)(28,44)(29,43)(31,35)(32,45)(33,46)(39,48)(41,47);(1,4,5,12)(2,8,7,14)(3,11,10,16)(6,13,9,15)(21,36)(27,43)(30,45)(34,35)(37,38)(40,42)(44,48)(46,47)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=g,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=cf,d^c=di,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, 6283800);