#147934

order	512
nilpotency class	4
isoclinism	isoclinic class of #13442
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2054
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11022
prime radical	C₂×#3414	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

955 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:

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

Polycyclic presentation:

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

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, 55130);