#6349132

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
prime radical	#62748	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2774 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)(3,11,12,15)(4,7,5,10)(8,13,16,14)(17,18,23,28)(19,29,32,42)(20,25,33,39)(21,27,22,26)(24,36,38,31)(30,43,44,48)(34,41,35,40)(37,46,47,45);(2,7)(4,5)(8,15)(9,10)(11,16)(13,14)(17,19)(18,24)(20,30)(21,31)(22,36)(23,32)(25,37)(26,29)(27,42)(28,38)(33,44)(34,45)(35,46)(39,47)(40,43)(41,48);(1,3)(2,8)(4,14)(5,13)(6,12)(7,15)(9,16)(10,11)(17,20)(18,25)(19,30)(21,34)(22,35)(23,33)(24,37)(26,40)(27,41)(28,39)(29,43)(31,45)(32,44)(36,46)(38,47)(42,48);(2,9)(3,12)(7,10)(13,14)(20,33)(25,39)(30,44)(34,35)(37,47)(40,41)(43,48)(45,46);(2,9)(7,10)(8,16)(11,15)(17,21,23,22)(18,26,28,27)(19,31,32,36)(20,34,33,35)(24,29,38,42)(25,40,39,41)(30,45,44,46)(37,43,47,48)

Polycyclic presentation:

2^9|a^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, 6256328);