#146915

order	512
nilpotency class	4
isoclinism	isoclinic class of #12026
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄²

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×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 492 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)(7,20)(8,26,29,42)(9,17)(10,15)(11,34)(12,19)(13,22)(14,28,41,60)(16,32)(18,31)(21,48,51,35)(23,46)(24,36)(25,47)(27,57,59,43)(30,61,58,40)(33,44)(37,53)(38,55)(39,54)(45,50,65,68)(49,56,67,62)(52,63,66,64);(1,2,3,4)(5,7,10,23,11,24,32,54,12,25,33,55,31,53,9,22)(6,13,17,37,18,38,44,47,19,39,16,36,34,46,15,20)(8,21,28,50,30,52,59,67,29,51,60,68,58,66,27,49)(14,35,42,62,43,63,61,65,41,48,26,56,57,64,40,45);(5,8,9,27,31,58,33,60,12,29,32,59,11,30,10,28)(6,14,15,40,34,57,16,26,19,41,44,61,18,43,17,42)(7,21,22,49,53,66,55,68,25,51,54,67,24,52,23,50)(13,35,20,45,46,64,36,56,39,48,47,65,38,63,37,62)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=d,d^2=hi,e^2=h,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, 54111);