#174315

order	512
nilpotency class	3
isoclinism	isoclinic class of #174093
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

773 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 773 conjugacy classes of subgroups — too many to draw. Only its 152 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)(11,15)(12,20)(13,17)(14,19)(16,22)(18,24)(21,23)(25,26,29,32)(27,33,35,40)(28,36,38,31)(30,37,39,34);(1,2)(3,6)(4,8)(5,7)(9,11)(10,14)(12,21)(13,18)(15,20)(16,24)(17,23)(19,22);(3,4)(6,8)(9,11)(10,15)(12,19)(13,18)(14,20)(16,23)(17,24)(21,22)(25,27,29,35)(26,30,32,39)(28,37,38,34)(31,40,36,33);(1,3,5,4)(2,6,7,8)(11,18)(14,23)(15,24)(19,21)

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=h,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dh,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

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 81511);