#305008

order	512
nilpotency class	3
isoclinism	isoclinic class of #268211
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 ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

657 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 657 conjugacy classes of subgroups — too many to draw. Only its 140 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:

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

Polycyclic presentation:

2^9|a^2=ei,b^2=eh,c^2=efh,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,d^c=dhi,g^c=gh

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, 212204);