#6811878

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806827
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	#3061	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1544 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=h,d^2=fi,e^2=fhi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,e^b=efh,h^b=hi,g^c=gi

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, 6719074);