#132104

order	512
nilpotency class	4
isoclinism	isoclinic class of #131988
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

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

normal subgroups only

This group has 636 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,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=ehi,f^b=fgh,d^c=dhi,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, 39300);