#132152

order	512
nilpotency class	4
isoclinism	isoclinic class of #131970
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1781
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄⋊C₄	C₂³
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#126 ◁ G
LECS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

629 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 629 conjugacy classes of subgroups — too many to draw. Only its 80 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) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2,6,12,7,14,23,47)(3,15,19,34,18,44,31,62)(4,10,24,41,25,43,30,11)(5,27,26,42,29,48,60,64)(8,13,32,45,22,39,57,40)(9,33,36,56,35,21,58,59)(16,46,28,37,53,49,17,54)(20,50,55,38,51,61,52,63);(1,3,8,20)(2,9,13,37)(4,21,26,16)(5,28,30,59)(6,31,32,52)(7,18,22,51)(10,38,42,34)(11,44,27,61)(12,46,45,58)(14,35,39,54)(15,48,50,41)(17,24,56,29)(19,57,55,23)(25,33,60,53)(36,47,49,40)(43,63,64,62);(1,4,22,60)(2,10,39,64)(3,16,51,33)(5,6,24,57)(7,25,8,26)(9,34,54,63)(11,45,48,47)(12,41,40,27)(13,42,14,43)(15,49,61,58)(17,55,59,31)(18,53,20,21)(19,28,52,56)(23,30,32,29)(35,62,37,38)(36,44,46,50)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=hi,d^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ci,d^b=dh,e^b=eh,f^b=fgh,d^c=dh,f^d=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, 39348);