#6481909

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470421
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	#29748	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1668 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)(4,7)(5,8)(9,13)(10,14)(11,15)(12,16)(18,23)(19,28)(21,24)(26,38)(27,39)(30,43)(31,32)(33,42)(34,35)(37,41)(45,46)(47,48);(1,3)(2,6)(4,11)(5,9)(7,15)(8,13)(10,12)(14,16)(17,18,24,39,22,27,21,23)(19,29,28,37,32,25,31,41)(20,33,34,38,36,26,35,42)(30,40,43,48,46,44,45,47);(3,9)(6,13)(10,11)(14,15)(17,19)(18,25)(20,30)(21,31)(22,32)(23,37)(24,28)(26,40)(27,29)(33,44)(34,43)(35,45)(36,46)(38,47)(39,41)(42,48);(1,4,5,12)(2,7,8,16)(3,10,9,11)(6,14,13,15)(17,20)(18,26)(19,30)(21,35)(22,36)(23,38)(24,34)(25,40)(27,33)(28,43)(29,44)(31,45)(32,46)(37,47)(39,42)(41,48);(2,8)(6,13)(7,16)(14,15)(17,21,22,24)(18,23,27,39)(19,31,32,28)(20,34,36,35)(25,37,29,41)(26,42,33,38)(30,43,46,45)(40,48,44,47)

Polycyclic presentation:

2^9|b^2=fi,d^2=hi,e^2=i,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,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, 6389105);