#146830

order	512
nilpotency class	3
isoclinism	isoclinic class of #11247
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#387
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10796
prime radical	C₂⁴	#131
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

447 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=d,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,g^a=gi,c^b=ch,d^b=di,e^c=ei,f^c=fi,g^c=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265425, #72265430, #72265455
maximal normal subgroup of	#72265425, #72265430, #72265455

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54026);