#140262

order	512
nilpotency class	4
isoclinism	isoclinic class of #12463
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1675
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#9567
prime radical	C₂³	#471
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

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

normal subgroups only

This group has 207 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47458);