#6394034

order	512
nilpotency class	3
isoclinism	isoclinic class of #62283
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

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	C₂³×C₂²⋊C₄
prime radical	#62752	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2808 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)(17,18,22,26,23,27,21,36)(19,24,31,39,29,38,30,45)(20,32,35,40,33,46,34,25)(28,41,44,47,42,48,43,37);(1,3)(2,7)(4,11)(5,13)(6,12)(8,15)(9,10)(14,16)(17,19)(18,24)(20,28)(21,30)(22,31)(23,29)(25,37)(26,39)(27,38)(32,41)(33,42)(34,43)(35,44)(36,45)(40,47)(46,48);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,15,10,16)(17,20)(18,25,27,40)(19,28)(21,34)(22,35)(23,33)(24,37,38,47)(26,32,36,46)(29,42)(30,43)(31,44)(39,41,45,48);(3,12)(7,10)(11,13)(15,16)(20,33)(25,40)(28,42)(32,46)(34,35)(37,47)(41,48)(43,44);(1,5,6,4)(2,8,9,14)(3,13,12,11)(7,15,10,16)(19,29)(24,38)(28,42)(30,31)(37,47)(39,45)(41,48)(43,44)

Polycyclic presentation:

2^9|a^2=h,c^2=fgh,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^b=ei,d^c=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303398
maximal normal subgroup of	#72303398

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6301230);