#147943

order	512
nilpotency class	4
isoclinism	isoclinic class of #13334
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2011
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

953 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:

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

Polycyclic presentation:

2^9|a^2=g,d^2=h,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=dhi,e^b=ei,f^b=fh,g^b=gi,h^b=hi,d^c=dh,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, 55139);