#244314

order	512
nilpotency class	3
isoclinism	isoclinic class of #233431
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

690 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 690 conjugacy classes of subgroups — too many to draw. Only its 186 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)(3,6,7,8)(9,10,12,16,13,17,22,15)(11,18,20,23,19,24,21,14)(25,26,28,33,30,34,29,31)(27,35,37,32,38,39,36,40);(2,5)(6,8)(9,11,13,19)(10,14,17,23)(12,21,22,20)(15,18,16,24)(26,31)(27,36)(28,29)(33,34)(35,39)(37,38);(10,15)(12,22)(14,18)(16,17)(20,21)(23,24)(25,27,30,38)(26,32,34,40)(28,36,29,37)(31,39,33,35);(1,3,4,7)(2,6,5,8)(11,19)(14,23)(18,24)(20,21)(25,28,30,29)(26,33,34,31)(27,36,38,37)(32,35,40,39)

Polycyclic presentation:

2^9|a^2=e,b^2=hi,c^2=i,d^2=efh,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,c^b=cg,d^b=dh,e^b=eh,f^b=fh,g^b=gi,e^c=eh,f^c=fh,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151510);