#6368782

order	512
nilpotency class	3
isoclinism	isoclinic class of #6368766
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	#3171	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1467 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6275978);