#6436625

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	#3449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2816 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)(17,18,23,20)(19,27,30,40)(21,32,34,25)(22,33,37,26)(24,36,31,29)(28,42,43,41)(35,39,47,46)(38,44,45,48);(1,3)(2,6)(4,10)(5,9)(7,12)(8,13)(11,15)(14,16)(17,19)(18,24)(20,31)(21,28)(22,36)(23,30)(25,38)(26,40)(27,33)(29,37)(32,45)(34,43)(35,48)(39,42)(41,46)(44,47);(3,9)(6,12)(10,15)(13,16)(17,20,23,18)(19,24,30,31)(21,32,34,25)(22,26,37,33)(27,29,40,36)(28,38,43,45)(35,39,47,46)(41,44,42,48);(2,7)(6,12)(8,14)(13,16)(17,21)(18,25)(19,28)(20,32)(22,35)(23,34)(24,38)(26,39)(27,41)(29,44)(30,43)(31,45)(33,46)(36,48)(37,47)(40,42);(1,4,5,11)(2,8,7,14)(3,10,9,15)(6,13,12,16)(21,34)(25,32)(28,43)(35,47)(38,45)(39,46)(41,42)(44,48)

Polycyclic presentation:

2^9|a^2=i,c^2=i,e^2=hi,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, 6343821);