#6350496

order	512
nilpotency class	3
isoclinism	isoclinic class of #6350397
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	C₂×2₋¹⁺⁴	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2683 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6257692);