#93685

order	512
nilpotency class	5
isoclinism	isoclinic class of #7348
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₆ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1317
commutator	C₂²×C₄	C₂×C₁₆
Frattini, cosocle	#2112	C₂²
socle	C₂	#7055
prime radical	C₂⁴	M₃₂
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #2112 ◁ G
LECS	G ▷ #2112 ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

145 conjugacy classes of subgroups; only the 32 normal subgroups are shown.

normal subgroups only

This group has 145 conjugacy classes of subgroups — too many to draw. Only its 32 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) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 2 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=g,c^2=i,d^2=f,f^2=h,h^2=i,b^a=bc,c^a=ce,e^a=eg,g^a=gi,d^b=dei,f^b=fi,d^c=dg,e^d=ei

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, 881);