#6362483

order	512
nilpotency class	3
isoclinism	isoclinic class of #6362318
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	C₄²	C₂⁵
prime radical	#61450	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1522 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6269679);