#132114

order	512
nilpotency class	4
isoclinism	isoclinic class of #131997
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	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	#51753	C₂
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#542 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

629 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 629 conjugacy classes of subgroups — too many to draw. Only its 78 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=h,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=dh,e^b=ehi,f^b=fgh,d^c=dhi,e^c=ei,f^c=fi,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, 39310);