#111470

order	512
nilpotency class	3
isoclinism	isoclinic class of #111467
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator, prime radical	C₂⁵	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

1100 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 1100 conjugacy classes of subgroups — too many to draw. Only its 142 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fh,c^b=cf,e^b=ehi,g^b=gh,d^c=di,g^c=gh

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, 18666);