#6362008

order	512
nilpotency class	3
isoclinism	isoclinic class of #6361928
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
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1534 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6269204);