#6369046

order	512
nilpotency class	3
isoclinism	isoclinic class of #6368982
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	#62324	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1512 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)(4,13)(5,7)(6,11)(8,14)(10,16)(12,15)(17,18)(19,24)(20,26)(21,28)(22,27)(23,29)(25,33)(30,38)(31,43)(32,40)(34,42)(35,39)(36,41)(37,45)(44,48)(46,47);(1,3,5,12)(2,6,7,16)(4,10,14,11)(8,15,13,9)(17,19,23,32)(18,24,29,40)(20,30,34,46)(21,35,22,31)(25,37,41,48)(26,38,42,47)(27,43,28,39)(33,45,36,44);(1,2)(3,6)(4,8)(5,7)(9,11)(10,15)(12,16)(13,14)(17,20)(18,25)(19,30)(21,33)(22,36)(23,34)(24,37)(26,27)(28,42)(29,41)(31,44)(32,46)(35,45)(38,43)(39,47)(40,48);(3,10)(4,14)(6,15)(8,13)(9,16)(11,12)(17,20)(18,26)(19,30)(21,36)(22,33)(23,34)(24,38)(25,27)(28,41)(29,42)(31,45)(32,46)(35,44)(37,43)(39,48)(40,47);(2,7)(6,16)(8,13)(9,15)(17,21,23,22)(18,27,29,28)(19,31,32,35)(20,33,34,36)(24,39,40,43)(25,26,41,42)(30,44,46,45)(37,47,48,38)

Polycyclic presentation:

2^9|b^2=hi,e^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fh,g^a=gi,d^b=df,e^b=ei,f^b=fh,e^c=eh,g^c=gi,e^d=ei,f^d=fh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6276242);