#309649

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	#2537	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 622 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)(3,6)(4,5)(7,8)(9,10)(11,17)(12,20)(13,15)(14,18)(16,22)(19,24)(21,23)(25,26)(27,30)(28,31)(29,32)(33,40)(34,38)(35,37)(36,39);(9,11,13,19)(10,14,15,23)(12,21,22,18)(16,24,20,17)(25,27,28,34)(26,30,31,38)(29,33,36,35)(32,40,39,37);(2,5)(6,8)(10,15)(14,23)(16,20)(17,24)(26,31)(27,33)(29,36)(30,37)(34,35)(38,40);(1,3,4,7)(2,6,5,8)(10,16)(12,22)(14,24)(15,20)(17,23)(18,21)(27,34)(30,38)(33,35)(37,40)

Polycyclic presentation:

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

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, 216845);