#312157

order	512
nilpotency class	3
isoclinism	isoclinic class of #244791
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

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

540 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 146 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)(9,10,13,18,14,19,24,17)(11,20,21,16,12,22,23,15)(27,33)(30,39)(34,40)(35,38);(2,5)(6,7)(9,11,14,12)(10,15,19,16)(13,23,24,21)(17,20,18,22)(25,26,28,31)(27,34,33,40)(29,37,36,32)(30,38,39,35);(9,12,14,11)(10,16,19,15)(13,21,24,23)(17,22,18,20)(25,27,28,33)(26,30,31,39)(29,38,36,35)(32,40,37,34);(1,3,4,8)(2,6,5,7)(10,17)(13,24)(15,20)(16,22)(18,19)(21,23)

Polycyclic presentation:

2^9|a^2=e,b^2=hi,c^2=hi,d^2=efhi,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,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

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 219353);