#312105

order	512
nilpotency class	3
isoclinism	isoclinic class of #244791
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

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

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

normal subgroups only

This group has 554 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)(9,10,15,18,14,20,13,19)(11,21,24,16,12,23,22,17)(25,26,28,31)(27,33,35,30)(29,32,37,40)(34,39,36,38);(2,5)(6,7)(9,11,14,12)(10,16,20,17)(13,24,15,22)(18,21,19,23)(25,27,28,35)(26,30,31,33)(29,34,37,36)(32,38,40,39);(9,12,14,11)(10,17,20,16)(13,22,15,24)(18,23,19,21)(27,34)(29,37)(30,38)(32,40)(33,39)(35,36);(1,3,4,8)(2,6,5,7)(10,18)(13,15)(16,21)(17,23)(19,20)(22,24)(26,31)(30,33)(32,40)(38,39)

Polycyclic presentation:

2^9|a^2=eh,b^2=hi,c^2=h,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, 219301);