#132459

order	512
nilpotency class	4
isoclinism	isoclinic class of #132279
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1782
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×Q₈	C₂³
prime radical	C₂²×D₄	C₂²⋊C₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #2356 ◁ G
LECS	G ▷ C₂×C₄×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁

457 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 457 conjugacy classes of subgroups — too many to draw. Only its 76 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=gh,d^2=h,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,g^a=gi,c^b=ci,d^b=dh,e^b=eh,f^b=fgh,d^c=dh,f^c=fg,f^d=fi,f^e=fi

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, 39655);