#141130

order	512
nilpotency class	4
isoclinism	isoclinic class of #125300
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1505
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

465 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 465 conjugacy classes of subgroups — too many to draw. Only its 96 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,c^2=ei,d^2=gh,e^2=i,f^2=gh,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,h^a=hi,d^b=dh,f^b=fhi,g^b=gi,f^d=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 48326);