#6362773

order	512
nilpotency class	3
isoclinism	isoclinic class of #6362750
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	#61731	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1587 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)(3,6)(4,7)(5,8)(9,12)(10,13)(11,14)(15,16)(17,18,22,25)(19,27,28,43)(20,31,34,24)(21,35,38,26)(23,36,39,30)(29,41,46,45)(32,40,47,48)(33,42,37,44);(1,3,5,11)(2,6,8,14)(4,10,12,16)(7,13,9,15)(17,19)(18,23)(20,32)(21,36)(22,28)(24,41)(25,39)(26,43)(27,35)(29,33)(30,38)(31,45)(34,47)(37,46)(40,42)(44,48);(2,7)(4,12)(6,13)(8,9)(10,16)(14,15)(19,28)(23,39)(27,43)(29,46)(30,36)(32,47)(40,48)(41,45);(17,20,22,34)(18,24,25,31)(19,29,28,46)(21,37,38,33)(23,40,39,48)(26,44,35,42)(27,45,43,41)(30,47,36,32);(2,8)(6,14)(7,9)(13,15)(18,25)(23,39)(24,31)(26,35)(27,43)(40,48)(41,45)(42,44)

Polycyclic presentation:

2^9|a^2=h,b^2=i,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,c^b=ch,d^b=df,f^b=fh,g^c=gi,f^d=fh

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6269969);