#6349133

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

2688 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:

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

Polycyclic presentation:

2^9|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,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	#72306694, #72306698, #72306740
maximal normal subgroup of	#72306694, #72306698, #72306740

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6256329);