#6349403

order	512
nilpotency class	3
isoclinism	isoclinic class of #6349370
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	#62827	C₂
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:

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

Polycyclic presentation:

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6256599);