#312282

order	512
nilpotency class	3
isoclinism	isoclinic class of #244927
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

604 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 604 conjugacy classes of subgroups — too many to draw. Only its 142 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:

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

Polycyclic presentation:

2^9|a^2=ehi,c^2=hi,d^2=efhi,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=deh,c^b=cg,d^b=di,e^b=eh,f^b=fh,g^b=gi,e^c=eh,f^c=fh,g^c=gi,e^d=eh,f^d=fh

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, 219478);