#297037

order	512
nilpotency class	3
isoclinism	isoclinic class of #268139
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

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 ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

675 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 675 conjugacy classes of subgroups — too many to draw. Only its 148 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=h,c^2=fhi,d^2=e,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=ghi,g^c=gh

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, 204233);