#6436619

order	512
nilpotency class	3
isoclinism	isoclinic class of #6432881
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	#29620	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2836 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)(3,6)(4,8)(5,7)(9,12)(10,13)(11,14)(15,16)(18,23)(21,32)(24,37)(25,36)(26,33)(28,34)(31,43)(35,39)(38,47)(40,44)(42,45)(46,48);(1,3)(2,6)(4,10)(5,9)(7,12)(8,13)(11,15)(14,16)(17,18)(19,24)(20,25)(21,33)(22,26)(23,32)(27,38)(28,35)(29,39)(30,40)(31,44)(34,37)(36,43)(41,48)(42,46)(45,47);(3,9)(6,12)(10,15)(13,16)(17,19,22,29)(18,24,26,39)(20,27,30,41)(21,34,32,28)(23,35,33,37)(25,38,40,48)(31,45,43,42)(36,46,44,47);(2,7)(6,12)(8,14)(13,16)(17,20)(18,25)(19,27)(21,31)(22,30)(23,36)(24,38)(26,40)(28,42)(29,41)(32,43)(33,44)(34,45)(35,46)(37,47)(39,48);(1,4,5,11)(2,8,7,14)(3,10,9,15)(6,13,12,16)(20,30)(25,40)(27,41)(31,43)(36,44)(38,48)(42,45)(46,47)

Polycyclic presentation:

2^9|c^2=i,e^2=h,f^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,c^b=ch,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, 6343815);