#6349204

order	512
nilpotency class	3
isoclinism	isoclinic class of #6349137
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2766 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)(3,10)(4,8)(5,13)(6,9)(7,11)(12,15)(14,16)(17,18)(19,29)(20,34)(21,27)(22,28)(23,26)(24,31)(25,35)(30,44)(32,39)(33,40)(36,41)(37,42)(38,47)(43,45)(46,48);(1,3,6,12)(2,7,9,16)(4,11,5,14)(8,15,13,10)(17,19,23,33)(18,24,26,39)(20,30,37,47)(21,32,22,31)(25,38,41,44)(27,29,28,40)(34,48,42,43)(35,46,36,45);(17,20)(18,25)(19,30)(21,35)(22,36)(23,37)(24,38)(26,41)(27,42)(28,34)(29,43)(31,45)(32,46)(33,47)(39,44)(40,48);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(18,26)(24,39)(25,41)(27,28)(29,40)(34,42)(38,44)(43,48);(3,12)(7,16)(10,15)(11,14)(17,21,23,22)(18,27,26,28)(19,31,33,32)(20,35,37,36)(24,40,39,29)(25,42,41,34)(30,45,47,46)(38,48,44,43)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306952, #72306963, #72306971, …
maximal normal subgroup of	#72306952, #72306963, #72306971, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6256400);