#6367118

order	512
nilpotency class	3
isoclinism	isoclinic class of #6366870
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	#3483	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1496 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=h,d^2=h,e^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,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, 6274314);