#303944

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	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2819	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 793 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=ef,d^2=ehi,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, 211140);