#241616

order	512
nilpotency class	3
isoclinism	isoclinic class of #233734
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	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

713 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 713 conjugacy classes of subgroups — too many to draw. Only its 138 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)(9,10,14,17)(11,15,21,22)(12,16,13,18)(19,23,20,24)(25,26)(27,35)(28,34)(29,33)(30,32)(31,37)(36,40)(38,39);(2,5)(6,8)(9,11,14,21)(10,15,17,22)(12,20,13,19)(16,24,18,23)(25,27,29,37)(26,31,33,35)(28,36,30,38)(32,40,34,39);(10,16)(11,19)(12,13)(17,18)(20,21)(23,24)(26,32)(28,30)(31,39)(33,34)(35,40)(36,38);(1,3,4,7)(2,6,5,8)(10,17)(15,22)(16,18)(23,24)(25,28,29,30)(26,32,33,34)(27,36,37,38)(31,40,35,39)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 148812);