#6349136

order	512
nilpotency class	3
isoclinism	isoclinic class of #6349099
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

2617 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)(3,11)(4,7)(5,9)(6,10)(8,13)(12,15)(14,16)(17,18)(19,24)(20,31)(21,27)(22,34)(23,26)(25,32)(28,43)(29,39)(30,38)(33,42)(35,40)(36,41)(37,46)(44,48)(45,47);(1,3)(2,8)(4,12)(5,13)(6,15)(7,14)(9,16)(10,11)(17,19)(18,24)(20,28)(21,29)(22,35)(23,30)(25,37)(26,38)(27,39)(31,47)(32,44)(33,45)(34,40)(36,46)(41,48)(42,43);(1,4,7,2)(3,12,14,8)(5,9,6,10)(11,13,16,15)(17,20)(18,25)(19,28)(21,32)(22,36)(23,33)(24,37)(26,41)(27,42)(29,44)(30,45)(31,34)(35,46)(38,48)(39,43)(40,47);(1,5,7,6)(2,9,4,10)(3,13,14,15)(8,16,12,11)(18,26)(24,38)(25,41)(27,34)(31,42)(37,48)(39,40)(43,47);(1,5,7,6)(2,10,4,9)(3,13,14,15)(8,11,12,16)(17,21,23,22)(18,27,26,34)(19,29,30,35)(20,32,33,36)(24,39,38,40)(25,42,41,31)(28,44,45,46)(37,43,48,47)

Polycyclic presentation:

2^9|c^2=h,d^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,f^b=fh,g^b=gi,d^c=dh,f^c=fi,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306633, #72306697, #72306723
maximal normal subgroup of	#72306633, #72306697, #72306723

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6256332);