#6362716

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

1569 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)(3,7)(4,12)(5,9)(6,10)(8,11)(13,15)(14,16)(17,18)(19,27)(20,24)(21,34)(22,25)(23,30)(26,35)(28,41)(29,37)(31,40)(32,45)(33,38)(36,43)(39,46)(42,47)(44,48);(1,3,6,5)(2,7,10,9)(4,11,13,14)(8,15,16,12)(17,19)(18,23)(20,28)(21,29)(22,31)(24,36)(25,37)(26,40)(27,34)(30,35)(32,42)(33,44)(38,47)(39,48)(41,45)(43,46);(1,4)(2,8)(3,11)(5,14)(6,13)(7,15)(9,12)(10,16)(17,20,22,33)(18,24,25,38)(19,28,31,44)(21,32,35,46)(23,36,37,47)(26,39,34,45)(27,41,40,48)(29,42,30,43);(1,3,6,5)(2,7,10,9)(4,11,13,14)(8,15,16,12)(18,25)(19,29)(21,35)(23,27)(24,38)(28,42)(30,31)(32,46)(36,41)(37,40)(43,44)(47,48);(1,3,6,5)(2,9,10,7)(4,11,13,14)(8,12,16,15)(18,25)(23,37)(24,38)(26,34)(27,40)(36,47)(39,45)(41,48)

Polycyclic presentation:

2^9|b^2=i,c^2=h,d^2=i,e^2=i,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, 6269912);