#6368623

order	512
nilpotency class	3
isoclinism	isoclinic class of #6368558
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

1515 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=i,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,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

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6275819);