#293794

order	512
nilpotency class	3
isoclinism	isoclinic class of #271259
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

745 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 745 conjugacy classes of subgroups — too many to draw. Only its 154 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)(3,7)(4,5)(6,8)(9,10)(11,18)(12,22)(13,17)(14,16)(15,21)(19,24)(20,23)(25,26,30,35)(27,31,36,32)(28,38,37,40)(29,34,39,33);(2,5)(6,7)(9,11,13,21)(10,15,17,18)(12,20,14,19)(16,23,22,24)(25,27,30,36)(26,31,35,32)(28,39,37,29)(33,40,34,38);(11,19)(12,14)(15,23)(16,22)(18,24)(20,21)(25,28,39,27,30,37,29,36)(26,32,34,40,35,31,33,38);(1,3,4,8)(2,6,5,7)(9,12,13,14)(10,16,17,22)(11,20,21,19)(15,23,18,24)(27,36)(28,37)(31,32)(38,40)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=fh,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,d^b=dh,g^b=gi,d^c=di,g^c=gi

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, 200990);