#331627

order	512
nilpotency class	3
isoclinism	isoclinic class of #331233
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₈

Aliases

No aliases for this group.

Subgroups

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

627 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 627 conjugacy classes of subgroups — too many to draw. Only its 158 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=eg,d^2=ehi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,d^b=dh,d^c=de,f^c=fh,g^c=gi

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, 238823);