#149600

order	512
nilpotency class	3
isoclinism	isoclinic class of #1462
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊C₄
commutator	C₂³	C₂²×C₁₆
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂²	C₂×C₂²⋊C₁₆
prime radical	C₂⁴	C₂×C₁₆
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

294 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 106 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) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56796);