#6669711

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	#3489	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1671 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)(19,24)(20,26)(21,25)(22,27)(23,28)(29,38)(30,37)(31,39)(32,40)(33,42)(34,44)(35,41)(36,43)(45,48)(46,47);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(17,19,23,32)(18,24,28,40)(20,33,34,45)(21,31,22,29)(25,39,27,38)(26,42,44,48)(30,36,46,35)(37,43,47,41);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(18,25)(19,29)(20,34)(21,22)(26,43)(27,28)(30,45)(31,32)(33,46)(37,47)(38,39)(41,44);(2,8)(6,9)(7,14)(13,15);(3,11)(6,9)(10,16)(13,15)(17,20,23,34)(18,26,28,44)(19,30,32,46)(21,35,22,36)(24,37,40,47)(25,41,27,43)(29,33,31,45)(38,42,39,48)

Polycyclic presentation:

2^9|b^2=fi,c^2=f,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, 6576907);