#328077

order	512
nilpotency class	3
isoclinism	isoclinic class of #327117
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

667 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)(19,22)(20,23)(21,25)(24,28)(26,30)(27,29)(31,32)(33,34,37,39)(35,43,46,48)(36,41,38,42)(40,44,47,45);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(17,19,21,26)(18,22,25,30)(20,27,28,32)(23,29,24,31)(34,39)(35,44)(36,38)(40,43)(45,46)(47,48);(1,4)(2,7)(3,10)(5,12)(6,13)(8,14)(9,15)(11,16)(18,23)(20,28)(22,29)(24,25)(27,32)(30,31)(33,35,37,46)(34,40,39,47)(36,44,38,45)(41,43,42,48);(4,12)(7,14)(10,16)(13,15)(19,26)(22,30)(27,32)(29,31)(33,36,37,38)(34,41,39,42)(35,45,46,44)(40,48,47,43)

Polycyclic presentation:

2^9|a^2=e,b^2=eh,c^2=i,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=dh,f^b=fi,g^b=gi,d^c=de,f^c=fh,g^c=gi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 235273);