#6351790

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	#60802	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2745 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)(3,9)(4,12)(5,7)(6,11)(8,13)(10,16)(14,15)(17,18)(19,29)(20,34)(21,26)(22,28)(23,27)(24,32)(25,36)(30,38)(31,39)(33,42)(35,40)(37,41)(43,45)(44,47)(46,48);(4,13)(8,12)(9,16)(11,14)(17,19)(18,24)(20,30)(21,31)(22,32)(23,33)(25,38)(26,29)(27,39)(28,42)(34,47)(35,44)(36,45)(37,46)(40,48)(41,43);(1,3)(2,6)(4,14)(5,10)(7,15)(8,16)(9,12)(11,13)(17,20,23,37)(18,25,27,40)(19,30,33,46)(21,36,22,35)(24,38,39,48)(26,41,28,34)(29,43,42,47)(31,45,32,44);(2,7)(6,15)(8,12)(9,16)(17,21,23,22)(18,26,27,28)(19,31,33,32)(20,35,37,36)(24,29,39,42)(25,34,40,41)(30,44,46,45)(38,47,48,43);(18,27)(24,39)(25,40)(26,28)(29,42)(34,41)(38,48)(43,47)

Polycyclic presentation:

2^9|c^2=h,d^2=h,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	#72279992, #72280141
maximal normal subgroup of	#72279992, #72280141

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6258986);