#147229

order	512
nilpotency class	4
isoclinism	isoclinic class of #13005
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1874
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10884
prime radical	C₂⁴	#137
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

519 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54425);