#241664

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	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

722 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 722 conjugacy classes of subgroups — too many to draw. Only its 140 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)(11,19)(12,17)(13,16)(14,18)(15,21)(20,24)(22,23)(25,26,28,32)(27,34,35,30)(29,31,37,33)(36,40,38,39);(2,5)(6,8)(9,11,12,21)(10,15,17,19)(13,22,14,20)(16,24,18,23)(25,27,28,35)(26,30,32,34)(29,38,37,36)(31,40,33,39);(10,16)(11,20)(13,14)(17,18)(21,22)(23,24)(26,31)(29,37)(30,39)(32,33)(34,40)(36,38);(1,3,4,7)(2,6,5,8)(11,21)(15,19)(20,22)(23,24)(27,35)(30,34)(36,38)(39,40)

Polycyclic presentation:

2^9|a^2=i,b^2=h,d^2=eh,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, 148860);