#174754

order	512
nilpotency class	3
isoclinism	isoclinic class of #174477
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 ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

716 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 130 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=gi,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dgh,e^a=eg,f^a=fi,d^b=dgh,e^b=eg,f^b=fh,d^c=dg,e^c=eh,f^c=fi

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, 81950);