#174159

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, prime radical	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

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

normal subgroups only

This group has 711 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) = 512

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

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=g,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, 81355);