#6489618

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470037
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	C₂²×C₄	C₂⁵
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2700 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)(3,7)(4,8)(5,9)(6,10)(11,14)(12,15)(13,16)(18,24)(21,22)(25,40)(26,39)(27,36)(28,38)(30,37)(32,42)(33,35)(41,48)(43,47)(44,45);(1,3,6,13)(2,7,10,16)(4,12,5,11)(8,15,9,14)(17,18,23,27)(19,28,31,25)(20,32,34,26)(21,24,22,36)(29,41,46,43)(30,40,37,38)(33,39,35,42)(44,48,45,47);(1,4,6,5)(2,8,10,9)(3,11,13,12)(7,14,16,15)(17,19,23,31)(18,25,27,28)(20,29,34,46)(21,30,22,37)(24,38,36,40)(26,41,32,43)(33,44,35,45)(39,47,42,48);(1,5,6,4)(2,9,10,8)(3,12,13,11)(7,15,16,14)(17,20)(18,26)(19,29)(21,35)(22,33)(23,34)(24,39)(25,41)(27,32)(28,43)(30,45)(31,46)(36,42)(37,44)(38,47)(40,48);(2,10)(7,16)(8,9)(14,15)(17,21,23,22)(18,24,27,36)(19,30,31,37)(20,33,34,35)(25,38,28,40)(26,42,32,39)(29,44,46,45)(41,48,43,47)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6396814);