#163239

order	512
nilpotency class	3
isoclinism	isoclinic class of #163012
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

778 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 778 conjugacy classes of subgroups — too many to draw. Only its 210 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) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=g,e^2=h,f^2=i,b^a=be,c^a=cgh,d^a=dgi,e^a=eg,c^b=ci,e^b=eh,f^b=fgh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 70435);