#174678

order	512
nilpotency class	3
isoclinism	isoclinic class of #174477
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

699 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 699 conjugacy classes of subgroups — too many to draw. Only its 144 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) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=gi,d^2=h,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, 81874);