#6673600

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667760
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	#3494	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1645 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)(19,30)(20,35)(21,27)(22,26)(23,29)(24,32)(25,37)(28,40)(31,46)(33,43)(34,47)(36,42)(38,44)(39,48)(41,45);(1,3,5,11)(2,6,7,9)(4,12,13,10)(8,16,15,14)(17,19)(18,24)(20,36)(21,37)(22,32)(23,33)(25,44)(26,30)(27,45)(28,43)(29,47)(31,39)(34,40)(35,48)(38,41)(42,46);(17,20,22,31)(18,25,26,45)(19,21,32,38)(23,41,40,37)(24,42,30,48)(27,47,44,43)(28,46,29,35)(33,36,34,39);(2,7)(6,9)(8,15)(14,16)(18,26)(24,30)(25,45)(27,44)(28,29)(35,46)(42,48)(43,47);(1,4,5,13)(2,8,7,15)(3,10,11,12)(6,14,9,16)(17,21,22,38)(18,27,26,44)(19,31,32,20)(23,39,40,36)(24,35,30,46)(25,43,45,47)(28,42,29,48)(33,41,34,37)

Polycyclic presentation:

2^9|b^2=fi,c^2=i,e^2=f,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,e^c=ei,g^c=gi,h^c=hi,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, 6580796);