#328173

order	512
nilpotency class	3
isoclinism	isoclinic class of #327021
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

707 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=ehi,c^2=hi,d^2=i,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,g^a=gi,c^b=cg,d^b=di,f^b=fi,g^b=gi,d^c=de,f^c=fh,g^c=gi

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, 235369);