#6349776

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

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

Polycyclic presentation:

2^9|b^2=i,d^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	#72306670, #72306684, #72306723
maximal normal subgroup of	#72306670, #72306684, #72306723

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6256972);