#93969

order	512
nilpotency class	4
isoclinism	isoclinic class of #1343
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₆ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#99
commutator	C₂²×C₄	C₂×C₁₆
Frattini, cosocle	C₂×#346	C₂²
socle	C₂²	#1314
prime radical	C₂³	C₂²⋊C₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×#346 ◁ G
LECS	G ▷ C₂×#346 ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

143 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 143 conjugacy classes of subgroups — too many to draw. Only its 52 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 — 2 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=f,c^2=f,d^2=g,g^2=h,h^2=i,b^a=bc,c^a=ce,e^a=ei,c^b=cf,d^b=def,d^c=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1165);