#6349801

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
prime radical	#62759	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2779 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)(3,11,14,15)(4,10,5,8)(7,12,16,13)(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)(2,7)(4,12)(5,13)(6,14)(8,15)(9,16)(10,11)(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);(2,8)(4,5)(7,15)(9,10)(11,16)(12,13)(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);(2,9)(7,16)(8,10)(11,15)(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);(1,4,6,5)(2,10,9,8)(3,12,14,13)(7,11,16,15)(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|a^2=i,d^2=hi,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, #72306671, #72306677, …
maximal normal subgroup of	#72306662, #72306671, #72306677, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6256997);