#177732

order	512
nilpotency class	3
isoclinism	isoclinic class of #177385
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	#37216	C₂
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

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

normal subgroups only

This group has 756 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) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 84928);