#6498498

order	512
nilpotency class	3
isoclinism	isoclinic class of #6473160
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#60801	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1684 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)(4,11)(5,7)(8,12)(9,13)(10,16)(14,15)(17,18)(19,29)(20,33)(21,26)(22,28)(23,27)(24,31)(25,37)(30,42)(32,40)(34,39)(35,44)(36,43)(38,41)(45,46)(47,48);(1,3)(2,6)(4,10)(5,9)(7,13)(8,14)(11,16)(12,15)(17,19)(18,24)(20,34)(21,38)(22,31)(23,32)(25,42)(26,29)(27,41)(28,40)(30,36)(33,47)(35,46)(37,48)(39,44)(43,45);(2,7)(3,9)(8,11)(10,15)(17,20,21,35,23,36,22,37)(18,25,28,43,27,44,26,33)(19,30,38,48,32,34,31,46)(24,39,40,47,41,42,29,45);(3,9)(6,13)(10,15)(14,16)(17,21,23,22)(18,26,27,28)(19,31,32,38)(20,35,36,37)(24,40,41,29)(25,33,44,43)(30,46,34,48)(39,47,42,45);(1,4,5,12)(2,8,7,11)(3,10,9,15)(6,14,13,16)(18,27)(20,36)(24,41)(26,28)(29,40)(30,34)(35,37)(46,48)

Polycyclic presentation:

2^9|c^2=gh,d^2=i,e^2=hi,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,e^c=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, 6405694);