#148906

order	512
nilpotency class	4
isoclinism	isoclinic class of #134701
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1809
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11232
prime radical	C₂⁴	#135
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

465 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)(3,14)(4,21,24,38)(5,26)(6,29)(7,12)(8,17)(9,23,34,22)(10,15)(11,36)(13,27)(16,45,47,60)(18,31)(19,52)(20,37)(25,58,56,55)(28,39)(30,59,33,48)(32,51)(35,44,54,57)(40,46,53,50)(41,43)(42,49)(61,64,63,62);(1,3,15,43,6,19,28,18,7,20,36,51,27,49,5,17)(2,8,26,42,13,32,11,37,12,31,39,52,29,41,10,14)(4,16,44,62,25,48,23,50,24,47,57,64,56,59,22,46)(9,33,55,63,35,60,38,40,34,30,58,61,54,45,21,53);(1,4,5,22,27,56,36,57,7,24,28,23,6,25,15,44)(2,9,10,21,29,54,39,58,12,34,11,38,13,35,26,55)(3,16,17,46,49,59,51,64,20,47,18,50,19,48,43,62)(8,30,14,40,41,60,52,63,31,33,37,53,32,45,42,61)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56102);