#6369195

order	512
nilpotency class	3
isoclinism	isoclinic class of #6369150
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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fh,g^a=gi,c^b=ci,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, 6276391);