#146916

order	512
nilpotency class	4
isoclinism	isoclinic class of #12026
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#427
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1808
prime radical	C₂⁴	C₂×Q₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

476 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 202 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) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 68 points:

(5,6,11,17,12,20,33,19)(7,21,25,37,26,50,57,39)(8,27,30,43,32,45,31,62)(9,18,10,15,34,16,35,47)(13,24,38,58,41,59,40,23)(14,42,44,29,46,28,48,61)(22,51,54,65,56,36,55,63)(49,66,64,53,67,52,60,68);(1,2,3,4)(5,7,10,24,11,25,34,58,12,26,35,59,33,57,9,23)(6,13,18,39,19,40,47,50,20,41,16,37,17,38,15,21)(8,22,29,53,31,55,42,66,32,56,61,68,30,54,28,52)(14,36,45,64,44,63,62,67,46,51,27,60,48,65,43,49);(5,8,9,28,33,30,35,61,12,32,34,42,11,31,10,29)(6,14,15,43,17,48,16,27,20,46,47,62,19,44,18,45)(7,22,23,52,57,54,59,68,26,56,58,66,25,55,24,53)(13,36,21,49,38,65,37,60,41,51,50,67,40,63,39,64)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=d,d^2=hi,e^2=h,f^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=ghi,h^a=hi,f^c=fi

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, 54112);