#132456

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×D₄	C₂³
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #2353 ◁ G
LECS	G ▷ C₂×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 477 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=gh,d^2=h,e^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, 39652);