#6673809

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667964
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

1633 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)(19,28)(20,33)(21,26)(22,38)(23,27)(24,31)(25,36)(29,42)(30,39)(32,41)(34,45)(35,44)(37,43)(40,48)(46,47);(1,3,5,11)(2,6,7,9)(4,12,13,10)(8,16,15,14)(17,19)(18,24)(20,34)(21,31)(22,32)(23,39)(25,42)(26,28)(27,41)(29,37)(30,38)(33,40)(35,48)(36,47)(43,46)(44,45);(17,20,21,35)(18,25,26,43)(19,29,31,47)(22,37,23,36)(24,40,28,45)(27,44,38,33)(30,46,41,42)(32,48,39,34);(2,7)(6,9)(8,15)(14,16)(18,26)(24,28)(25,43)(27,38)(30,41)(33,44)(40,45)(42,46);(1,4,5,13)(2,8,7,15)(3,10,11,12)(6,14,9,16)(17,18)(19,30)(20,25)(21,26)(22,27)(23,38)(24,32)(28,39)(29,46)(31,41)(33,36)(34,45)(35,43)(37,44)(40,48)(42,47)

Polycyclic presentation:

2^9|b^2=fi,c^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,e^d=ei,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, 6581005);