#6673591

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667748
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	#60580	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1648 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)(3,9)(4,8)(5,7)(6,11)(10,16)(12,14)(13,15)(17,18)(19,29)(20,35)(21,28)(22,27)(23,26)(24,33)(25,37)(30,46)(31,41)(32,34)(36,40)(38,44)(39,43)(42,48)(45,47);(1,3,5,11)(2,6,7,9)(4,12,13,10)(8,16,15,14)(17,19)(18,24)(20,36)(21,31)(22,33)(23,34)(25,42)(26,41)(27,29)(28,32)(30,37)(35,45)(38,47)(39,48)(40,44)(43,46);(17,20)(18,25)(19,30)(21,37)(22,38)(23,39)(24,40)(26,35)(27,43)(28,44)(29,45)(31,47)(32,46)(33,48)(34,36)(41,42);(2,7)(6,9)(8,15)(14,16)(17,21,22,23)(18,26,27,28)(19,31,33,34)(20,37,38,39)(24,41,29,32)(25,35,43,44)(30,47,48,36)(40,42,45,46);(1,4,5,13)(2,8,7,15)(3,10,11,12)(6,14,9,16)(17,18)(19,32)(20,25)(21,26)(22,27)(23,28)(24,31)(29,34)(30,46)(33,41)(35,37)(36,45)(38,43)(39,44)(40,47)(42,48)

Polycyclic presentation:

2^9|b^2=fi,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,h^e=hi

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, 6580787);