#244369

order	512
nilpotency class	3
isoclinism	isoclinic class of #233469
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

674 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 674 conjugacy classes of subgroups — too many to draw. Only its 162 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)(9,10,12,15)(11,17,19,14)(13,16,21,24)(18,23,20,22)(25,26,29,33,30,34,28,31)(27,35,36,39,37,40,38,32);(2,5)(6,8)(9,11,12,19)(10,14,15,17)(13,18,21,20)(16,22,24,23)(26,31)(27,36)(28,29)(32,39)(33,34)(37,38);(11,18)(13,21)(14,22)(16,24)(17,23)(19,20)(25,27,30,37)(26,32,34,39)(28,36,29,38)(31,35,33,40);(1,3,4,7)(2,6,5,8)(10,15)(14,17)(16,24)(22,23)(25,28,30,29)(26,33,34,31)(27,36,37,38)(32,40,39,35)

Polycyclic presentation:

2^9|a^2=e,b^2=hi,c^2=h,d^2=efhi,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,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

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151565);