#133673

order	512
nilpotency class	4
isoclinism	isoclinic class of #133329
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1790
commutator, prime radical	C₂²×D₄	C₂²×C₄
Frattini, cosocle	C₂×C₄×D₄	C₂³
LCS	G ▷ C₂²×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

517 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 517 conjugacy classes of subgroups — too many to draw. Only its 78 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) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2,5,10,6,9,4,8)(3,11,12,16,13,7,14,15)(17,18)(19,23)(20,30)(21,34)(22,26)(24,33)(25,36)(27,44)(28,42)(29,39)(31,48)(32,43)(35,41)(37,45)(38,47)(40,46);(1,3)(2,7)(4,12)(5,14)(6,13)(8,15)(9,11)(10,16)(17,19,22,29)(18,23,26,39)(20,31,32,27)(21,28,36,47)(24,40,41,37)(25,42,34,38)(30,44,43,48)(33,46,35,45);(1,4,6,5)(2,8,9,10)(3,12,13,14)(7,15,11,16)(17,20,22,32)(18,24,26,41)(19,27,29,31)(21,35,36,33)(23,37,39,40)(25,43,34,30)(28,46,47,45)(38,48,42,44)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=gi,e^2=i,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=ei,h^a=hi,c^b=ci,e^b=ehi,f^b=fg,d^c=dh,e^c=ei,e^d=ei

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 40869);