#6362771

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	#61737	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1590 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)(19,27,31,42)(20,24,34,40)(21,35,36,26)(23,29,38,30)(28,43,32,46)(33,47,48,41)(37,44,39,45);(1,3,5,11)(2,6,8,14)(4,10,12,16)(7,13,9,15)(17,19,22,31)(18,23,25,38)(20,32,34,28)(21,29,36,30)(24,39,40,37)(26,42,35,27)(33,45,48,44)(41,43,47,46);(2,7)(4,12)(6,13)(8,9)(10,16)(14,15)(17,20,22,34)(18,24,25,40)(19,28,31,32)(21,33,36,48)(23,37,38,39)(26,41,35,47)(27,43,42,46)(29,44,30,45);(18,25)(19,29)(21,36)(23,27)(24,40)(28,44)(30,31)(32,45)(33,48)(37,43)(38,42)(39,46);(2,8)(6,14)(7,9)(13,15)(18,25)(23,38)(24,40)(26,35)(27,42)(37,39)(41,47)(43,46)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^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, 6269967);