#6669712

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	#3488	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1692 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)(19,29)(20,27)(21,26)(22,25)(23,28)(24,32)(30,40)(31,45)(33,39)(34,48)(35,42)(36,41)(37,44)(38,47)(43,46);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(17,19)(18,24)(20,34)(21,32)(22,33)(23,30)(25,40)(26,29)(27,43)(28,39)(31,37)(35,46)(36,47)(38,44)(41,45)(42,48);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(18,25)(19,30)(20,35)(22,23)(26,28)(27,44)(31,34)(32,33)(38,45)(39,40)(41,42)(46,47);(2,8)(6,9)(7,14)(13,15)(18,26)(24,29)(25,28)(27,42)(38,45)(39,40)(41,44)(43,48);(3,11)(6,9)(10,16)(13,15)(17,20)(18,27)(19,31)(21,35)(22,36)(23,37)(24,38)(25,41)(26,42)(28,44)(29,45)(30,46)(32,47)(33,34)(39,48)(40,43)

Polycyclic presentation:

2^9|b^2=fi,c^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,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, 6576908);