#6366805

order	512
nilpotency class	3
isoclinism	isoclinic class of #6366710
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

1509 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)(3,11)(4,5)(8,12)(9,10)(15,16)(17,18,24,30)(19,31,36,42)(20,26,21,27)(22,29,23,28)(25,34,43,35)(32,47,37,41)(33,44,48,40)(38,46,39,45);(1,2,6,10)(3,12,11,15)(4,9,5,7)(8,13,16,14)(17,19,24,36)(18,25,30,43)(20,37,21,32)(22,35,23,34)(26,44,27,40)(28,31,29,42)(33,38,48,39)(41,45,47,46);(1,2)(3,8)(4,7)(5,9)(6,10)(11,16)(12,14)(13,15)(17,20,24,21)(18,26,30,27)(19,32,36,37)(22,39,23,38)(25,40,43,44)(28,46,29,45)(31,47,42,41)(33,35,48,34);(1,3,6,11)(2,8,10,16)(4,13,5,14)(7,15,9,12)(17,21,24,20)(18,27,30,26)(19,33,36,48)(22,39,23,38)(25,41,43,47)(28,46,29,45)(31,44,42,40)(32,35,37,34);(1,4,6,5)(2,9,10,7)(3,13,11,14)(8,12,16,15)(17,22,24,23)(18,28,30,29)(19,34,36,35)(20,38,21,39)(25,42,43,31)(26,45,27,46)(32,48,37,33)(40,47,44,41)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283673
maximal normal subgroup of	#72283673

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6274001);