#132492

order	512
nilpotency class	4
isoclinism	isoclinic class of #132309
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	C₂³	#379
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #2356 ◁ G
LECS	G ▷ C₂×C₄×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 443 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)(17,18,22,28,23,29,20,26)(19,30,31,24,33,40,37,41)(21,25,34,42,36,27,38,43)(32,45,48,39,35,46,47,44);(1,2)(3,7)(4,11)(5,8)(6,13)(9,14)(10,16)(12,15)(17,19,23,33)(18,24,29,41)(20,31,22,37)(21,35,36,32)(25,39,27,44)(26,30,28,40)(34,48,38,47)(42,46,43,45);(1,3,5,10)(2,7,8,16)(4,12,13,9)(6,14,11,15)(18,25,29,27)(20,22)(24,39,41,44)(26,43,28,42)(30,45,40,46)(31,37)(34,38)(47,48)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=ghi,d^2=h,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,g^a=gi,d^b=dh,e^b=ehi,f^b=fgh,d^c=dhi,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, 39688);