#6498493

order	512
nilpotency class	3
isoclinism	isoclinic class of #6473158
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)(25,35)(30,42)(32,40)(34,39)(36,44)(37,43)(38,41)(45,47)(46,48);(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)(30,37)(33,48)(35,46)(36,47)(39,44)(43,45);(2,7)(3,9)(8,11)(10,15)(17,20,21,36,23,37,22,35)(18,25,28,43,27,44,26,33)(19,30,38,46,32,34,31,47)(24,39,40,48,41,42,29,45);(3,9)(6,13)(10,15)(14,16)(17,21,23,22)(18,26,27,28)(19,31,32,38)(20,35,37,36)(24,40,41,29)(25,43,44,33)(30,46,34,47)(39,45,42,48);(1,4,5,12)(2,8,7,11)(3,10,9,15)(6,14,13,16)(18,27)(24,41)(25,44)(26,28)(29,40)(33,43)(39,42)(45,48)

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,d^c=di

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, 6405689);