#6674455

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667964
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#60765	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1737 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)(3,11)(4,8)(5,10)(6,9)(7,13)(12,16)(14,15)(17,18)(19,25)(20,35)(21,27)(22,28)(23,29)(24,30)(26,37)(31,42)(32,43)(33,45)(34,44)(36,47)(38,46)(39,41)(40,48);(1,3)(2,7)(4,12)(5,14)(6,13)(8,15)(9,11)(10,16)(17,19,24,34)(18,25,30,44)(20,36,32,39)(21,31,23,33)(22,38,40,37)(26,28,46,48)(27,42,29,45)(35,47,43,41);(1,4,6,5)(2,8,9,10)(3,12,13,14)(7,15,11,16)(17,20)(18,26)(19,22)(21,37)(23,38)(24,32)(25,41)(27,43)(28,45)(29,35)(30,46)(31,39)(33,36)(34,40)(42,48)(44,47);(2,9)(7,11)(8,10)(15,16)(17,21,24,23)(18,27,30,29)(19,31,34,33)(20,37,32,38)(22,39,40,36)(25,42,44,45)(26,43,46,35)(28,41,48,47);(1,5,6,4)(2,10,9,8)(3,12,13,14)(7,15,11,16)(17,22,24,40)(18,28,30,48)(19,32,34,20)(21,36,23,39)(25,43,44,35)(26,45,46,42)(27,47,29,41)(31,37,33,38)

Polycyclic presentation:

2^9|b^2=i,c^2=f,d^2=i,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,g^c=gi,h^c=hi,e^d=ei,h^e=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6581651);