#147700

order	512
nilpotency class	4
isoclinism	isoclinic class of #13402
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2034
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10969
prime radical	C₂×#3067	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1001 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:

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

Polycyclic presentation:

2^9|a^2=g,d^2=hi,e^2=i,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,h^a=hi,c^b=cf,d^b=dh,e^b=ei,f^b=fi,h^b=hi,e^c=ei,f^c=fi,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, 54896);