#6366685

order	512
nilpotency class	3
isoclinism	isoclinic class of #6366542
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

1536 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^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,d^b=df,e^b=ehi,f^b=fh,e^c=eh,g^c=gi,f^d=fh

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6273881);