#138467

order	512
nilpotency class	4
isoclinism	isoclinic class of #138406
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1874
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#447
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

335 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 335 conjugacy classes of subgroups — too many to draw. Only its 88 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)(3,11)(4,7)(6,9,14,16)(8,12)(13,15)(17,18,21,26,20,25,33,40,23,28,36,43,22,27,35,42)(19,29,31,44,37,48,47,39,32,45,46,38,30,41,34,24);(1,3,5,4)(2,7)(6,13,14,12)(8,9)(10,11)(15,16)(17,19,23,32)(18,24,28,39)(20,30,22,37)(21,34,36,47)(25,38,27,44)(26,41,43,48)(29,40,45,42)(31,33,46,35);(1,4,5,3)(2,8,10,15)(6,12,14,13)(7,9,11,16)(17,20,23,22)(18,25,28,27)(19,30,32,37)(21,33,36,35)(24,38,39,44)(26,40,43,42)(29,41,45,48)(31,34,46,47)

Polycyclic presentation:

2^9|a^2=f,b^2=dfhi,c^2=gi,d^2=ghi,e^2=g,f^2=ghi,h^2=i,b^a=bd,c^a=ce,d^a=dg,d^b=dh,e^b=eg,f^b=fh,h^b=hi,d^c=dg,e^c=eg,f^c=fg

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45663);