#140230

order	512
nilpotency class	4
isoclinism	isoclinic class of #12436
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1667
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#9563
prime radical	#3147	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

250 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 82 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) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=d,d^2=gi,e^2=h,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,e^c=egh,h^c=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225389, #72225396, #72225403
maximal normal subgroup of	#72225389, #72225396, #72225403
maximal quotient of	#72225530

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47426);