#6669709

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	#61174	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1667 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)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18)(19,23)(20,31)(21,29)(22,27)(24,26)(25,36)(28,33)(30,40)(32,35)(34,45)(37,38)(39,46)(41,43)(42,48)(44,47);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(17,19,23,18)(20,32,35,31)(21,29,24,26)(22,38,40,43)(25,33,28,36)(27,41,30,37)(34,42,39,44)(45,47,46,48);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,20,23,35)(18,25,19,28)(21,33,24,36)(22,39,40,34)(26,32,29,31)(27,44,30,42)(37,48,41,47)(38,46,43,45);(2,8)(6,9)(7,14)(13,15)(17,21,23,24)(18,26,19,29)(20,33,35,36)(22,37,40,41)(25,32,28,31)(27,43,30,38)(34,47,39,48)(42,46,44,45);(3,11)(6,9)(10,16)(13,15)(17,22,23,40)(18,27,19,30)(20,34,35,39)(21,37,24,41)(25,42,28,44)(26,43,29,38)(31,45,32,46)(33,47,36,48)

Polycyclic presentation:

2^9|b^2=f,c^2=f,d^2=i,e^2=i,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, 6576905);