#122601

order	512
nilpotency class	4
isoclinism	isoclinic class of #122554
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#327
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄⋊C₈	C₂³
prime radical	C₂⁴	#102
LCS	G ▷ C₂²×C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₄⋊C₈ ◁ G
LECS	G ▷ C₂×C₄⋊C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁

487 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 487 conjugacy classes of subgroups — too many to draw. Only its 80 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) = 64

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=i,d^2=gi,e^2=h,f^2=i,g^2=i,b^a=bd,d^a=df,f^a=fh,g^a=gi,c^b=ci,d^b=dg,e^b=efgi,f^b=fi,g^b=gi,e^d=ehi

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, 29797);