#135775

order	512
nilpotency class	4
isoclinism	isoclinic class of #135766
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1846
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#495
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

234 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 122 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) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 72 points:

(2,3)(4,5)(7,8)(9,10)(11,21,25,42)(12,27,29,17)(13,31)(14,33)(15,19)(16,24,37,45)(18,32)(20,34)(22,44)(23,48)(26,54,53,43)(28,58,46,61)(30,40,60,57)(35,47)(36,49)(38,64,62,52)(39,59,56,67)(41,51)(50,65)(55,66)(63,72)(68,70)(69,71);(1,2,5,7,6,8,4,3)(9,11,22,45,14,26,47,64,15,25,13,24,34,53,32,52)(10,16,35,42,20,38,44,54,19,37,18,21,33,62,31,43)(12,23,46,66,30,50,67,72,29,51,28,49,60,71,59,70)(17,36,56,48,40,63,58,69,27,55,39,41,57,68,61,65);(1,3,4,8,6,7,5,2)(9,12,14,30,15,29,34,60)(10,17,33,57,19,27,20,40)(11,23,26,50,25,51,53,71)(13,28,32,59,22,46,47,67)(16,36,62,68,37,55,38,63)(18,39,44,58,35,56,31,61)(21,41,54,69,42,48,43,65)(24,49,52,70,45,66,64,72)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42971);