#290947

order	512
nilpotency class	3
isoclinism	isoclinic class of #268211
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

678 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 678 conjugacy classes of subgroups — too many to draw. Only its 136 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 56 points:

(1,2)(3,7)(4,6)(5,8)(9,10,12,14)(11,17,18,24)(13,21,20,16)(15,23,22,19)(25,26)(27,32)(28,40)(29,34)(30,35)(31,36)(33,43)(37,50)(38,47)(39,49)(41,55)(42,52)(44,51)(45,48)(46,53)(54,56);(1,3,4,8)(2,5,6,7)(10,14)(11,18)(13,20)(15,22)(25,27,31,39)(26,32,36,49)(28,41,44,53)(29,38,30,45)(33,50,52,56)(34,47,35,48)(37,42,54,43)(40,55,51,46);(9,11,13,19,12,18,20,23)(10,15,16,17,14,22,21,24)(25,28,29,43,31,44,30,42)(26,33,35,40,36,52,34,51)(27,37,38,53,39,54,45,41)(32,46,48,50,49,55,47,56);(2,6)(5,7)(10,14)(15,22)(16,21)(17,24)(25,29,31,30)(26,34,36,35)(27,38,39,45)(28,42,44,43)(32,47,49,48)(33,40,52,51)(37,41,54,53)(46,50,55,56)

Polycyclic presentation:

2^9|a^2=i,b^2=eh,c^2=f,d^2=hi,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=ghi,d^c=dhi,g^c=gh

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 198143);