#6496122

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	#62847	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=i,d^2=i,e^2=h,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, 6403318);