#309584

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

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

normal subgroups only

This group has 532 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)(3,6)(4,5)(7,8)(9,10,14,19)(11,20,12,16)(13,17,23,18)(15,22,24,21)(25,26,29,30)(27,31,34,38)(28,35,37,32)(33,36,40,39);(9,11,14,12)(10,15,19,24)(13,22,23,21)(16,17,20,18)(25,27,29,34)(26,31,30,38)(28,36,37,39)(32,33,35,40);(2,5)(6,8)(9,12,14,11)(10,16,19,20)(13,22,23,21)(15,17,24,18)(25,28,29,37)(26,32,30,35)(27,33,34,40)(31,39,38,36);(1,3,4,7)(2,6,5,8)(10,17)(13,23)(15,20)(16,24)(18,19)(21,22)

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=hi,d^2=fi,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,e^b=eh,g^b=gi,g^c=gi,e^d=eh

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, 216780);