#6477433

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470067
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 ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2844 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,f^b=fi,g^c=gi,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, 6384629);