#6362563

order	512
nilpotency class	3
isoclinism	isoclinic class of #6362510
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	#121	C₂×D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1495 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)(3,9)(4,12)(5,7)(6,10)(8,14)(11,16)(13,15)(17,18,23,27)(19,28,31,39)(20,25,22,36)(21,33,35,26)(24,37,41,29)(30,43,47,45)(32,44,38,42)(34,40,46,48);(17,19,23,31)(18,24,27,41)(20,29,22,37)(21,34,35,46)(25,39,36,28)(26,43,33,45)(30,38,47,32)(40,42,48,44);(1,3,5,11)(2,6,7,15)(4,13,14,10)(8,16,12,9)(17,20,23,22)(18,25,27,36)(19,29,31,37)(21,32,35,38)(24,39,41,28)(26,42,33,44)(30,46,47,34)(40,43,48,45);(17,21,23,35)(18,26,27,33)(19,30,31,47)(20,32,22,38)(24,40,41,48)(25,42,36,44)(28,45,39,43)(29,46,37,34);(2,7)(6,15)(8,12)(9,16)(18,27)(24,41)(25,36)(26,33)(28,39)(40,48)(42,44)(43,45)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6269759);