#114928

order	512
nilpotency class	3
isoclinism	isoclinic class of #10078
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	#120
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

970 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 970 conjugacy classes of subgroups — too many to draw. Only its 186 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 — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,d^2=i,e^2=h,f^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fh,c^b=cf,d^b=di,e^b=ehi,f^b=fi,g^b=ghi,d^c=di,e^c=eh,f^c=fi,g^c=gh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 22124);