#172052

order	512
nilpotency class	3
isoclinism	isoclinic class of #171725
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

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

normal subgroups only

This group has 730 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 79248);