#152846

order	512
nilpotency class	2
isoclinism	isoclinic class of #392
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₁₆	C₂³
commutator, socle, prime radical	C₂³	C₂²×C₁₆
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

214 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 122 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) = 512

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=e,f^2=g,g^2=h,h^2=i,b^a=bd,c^a=ce,c^b=ci

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60042);