#96769

order	512
nilpotency class	3
isoclinism	isoclinic class of #7606
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	C₄×C₂²⋊C₄
prime radical	C₂⁴	#117
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

476 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 130 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=eh,d^2=g,e^2=g,f^2=i,b^a=bd,d^a=dg,f^a=fgh,c^b=chi,d^b=dh

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3965);