#6368406

order	512
nilpotency class	3
isoclinism	isoclinic class of #6368046
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	#62466	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1517 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)(3,4)(5,13)(6,9)(7,12)(8,11)(10,16)(14,15)(17,18,23,27)(19,24,30,39)(20,32,34,40)(21,26,22,35)(25,36,41,33)(28,42,44,47)(29,38,31,45)(37,46,48,43);(1,3)(2,7)(4,13)(5,15)(6,11)(8,10)(9,14)(12,16)(17,19,23,30)(18,24,27,39)(20,28,34,44)(21,29,22,31)(25,37,41,48)(26,38,35,45)(32,42,40,47)(33,43,36,46);(1,3)(2,8)(4,9)(5,12)(6,11)(7,10)(13,14)(15,16)(17,20)(18,25)(19,28)(21,33)(22,36)(23,34)(24,37)(26,40)(27,41)(29,43)(30,44)(31,46)(32,35)(38,47)(39,48)(42,45);(1,4,6,8)(2,3,9,11)(5,7,16,14)(10,15,13,12)(17,21,23,22)(18,26,27,35)(19,29,30,31)(20,33,34,36)(24,38,39,45)(25,40,41,32)(28,43,44,46)(37,47,48,42);(2,9)(3,11)(10,13)(12,15)(18,27)(19,30)(25,41)(26,35)(28,44)(29,31)(32,40)(43,46)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6275602);