#6351810

order	512
nilpotency class	3
isoclinism	isoclinic class of #6351787
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

2795 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)(3,11,14,16)(4,8,5,10)(7,12,15,13)(17,18,23,28)(19,29,32,42)(20,25,35,41)(21,27,22,26)(24,36,38,31)(30,44,47,43)(33,40,34,39)(37,46,48,45);(1,3)(2,7)(4,12)(5,13)(6,14)(8,11)(9,15)(10,16)(17,19)(18,24)(20,30)(21,31)(22,36)(23,32)(25,37)(26,29)(27,42)(28,38)(33,45)(34,46)(35,47)(39,44)(40,43)(41,48);(1,3)(2,7)(4,13)(5,12)(6,14)(8,16)(9,15)(10,11)(17,20)(18,25)(19,30)(21,34)(22,33)(23,35)(24,37)(26,40)(27,39)(28,41)(29,43)(31,46)(32,47)(36,45)(38,48)(42,44);(1,4,6,5)(2,8,9,10)(3,12,14,13)(7,11,15,16)(17,21,23,22)(18,26,28,27)(19,31,32,36)(20,33,35,34)(24,29,38,42)(25,39,41,40)(30,45,47,46)(37,44,48,43);(2,9)(7,15)(8,10)(11,16)(17,21,23,22)(18,26,28,27)(19,31,32,36)(20,34,35,33)(24,29,38,42)(25,40,41,39)(30,46,47,45)(37,43,48,44)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280141
maximal normal subgroup of	#72280141

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6259006);