#312280

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

582 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 138 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)(9,10,13,15,14,18,12,17)(11,19,21,24,22,23,20,16)(25,26,29,33)(27,31,30,34)(28,36,37,32)(35,40,38,39);(2,5)(6,7)(10,15)(12,13)(16,23)(17,18)(19,24)(20,21)(25,27)(26,31)(28,37)(29,30)(32,36)(33,34);(9,11,14,22)(10,16,18,24)(12,21,13,20)(15,23,17,19)(25,28,29,37)(26,32,33,36)(27,35,30,38)(31,39,34,40);(1,3,4,8)(2,6,5,7)(10,17)(12,13)(15,18)(16,19)(20,21)(23,24)(25,27,29,30)(26,31,33,34)(28,35,37,38)(32,39,36,40)

Polycyclic presentation:

2^9|a^2=ehi,c^2=hi,d^2=ef,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, 219476);