#136141

order	512
nilpotency class	4
isoclinism	isoclinic class of #12114
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1595
commutator	C₂×M₁₆	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
socle	C₂	#8866
prime radical	#3054	C₂²
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 384 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,d^2=gi,e^2=g,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,g^a=gi,d^b=dg,e^b=ehi,g^b=gi,h^b=hi,d^c=dh,e^c=egi,g^c=gi,h^c=hi,e^d=ei

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 43337);