#140798

order	512
nilpotency class	3
isoclinism	isoclinic class of #9784
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#390
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1954
prime radical	C₂²×D₄	C₂×C₈
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

297 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 297 conjugacy classes of subgroups — too many to draw. Only its 94 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) = 128

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=e,e^2=gi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,c^b=ci,d^b=dg,f^b=fg

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47994);