#6544042

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533075
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	#2006	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2687 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:

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

Polycyclic presentation:

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

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, 6451238);