#241261

order	512
nilpotency class	3
isoclinism	isoclinic class of #233668
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

754 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 754 conjugacy classes of subgroups — too many to draw. Only its 202 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:

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

Polycyclic presentation:

2^9|a^2=ei,b^2=e,c^2=e,f^2=h,g^2=hi,b^a=be,c^a=cf,f^a=fh,g^a=gi,c^b=cg,d^b=dh,f^b=fi,g^b=ghi,f^c=fh,g^c=ghi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 148457);