#6349782

order	512
nilpotency class	3
isoclinism	isoclinic class of #6349761
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

2618 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)(3,12)(4,14)(5,10)(6,9)(7,11)(8,15)(13,16)(17,18)(19,28)(20,25)(21,26)(22,35)(23,27)(24,30)(29,43)(31,38)(32,40)(33,41)(34,42)(36,39)(37,44)(45,48)(46,47);(1,3,7,8)(2,4,11,16)(5,13,6,14)(9,15,10,12)(17,19)(18,24)(20,29)(21,30)(22,36)(23,31)(25,37)(26,38)(27,39)(28,35)(32,44)(33,45)(34,46)(40,47)(41,43)(42,48);(1,4)(2,8)(3,11)(5,15)(6,12)(7,16)(9,14)(10,13)(17,20)(18,25)(19,29)(21,32)(22,33)(23,34)(24,37)(26,40)(27,42)(28,43)(30,44)(31,46)(35,41)(36,45)(38,47)(39,48);(1,5,7,6)(2,9,11,10)(3,13,8,14)(4,15,16,12)(17,21,23,22)(18,26,27,35)(19,30,31,36)(20,32,34,33)(24,38,39,28)(25,40,42,41)(29,44,46,45)(37,47,48,43);(1,5,7,6)(2,10,11,9)(3,13,8,14)(4,12,16,15)(17,21,23,22)(18,26,27,35)(19,30,31,36)(20,33,34,32)(24,38,39,28)(25,41,42,40)(29,45,46,44)(37,43,48,47)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306662, #72306666, #72306667, …
maximal normal subgroup of	#72306662, #72306666, #72306667, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6256978);