#146205

order	512
nilpotency class	4
isoclinism	isoclinic class of #146204
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2094
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#3169	C₂²
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1744 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

235 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 235 conjugacy classes of subgroups — too many to draw. Only its 64 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:

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

Polycyclic presentation:

2^9|a^2=dgi,b^2=de,d^2=g,e^2=hi,f^2=g,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=ei,f^a=fgh,c^b=cf,d^b=dg,e^b=egi,f^b=fg,g^b=gi,h^b=hi,d^c=dh,e^c=eh,f^c=fgi,g^c=gi,h^c=hi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53401);