#147685

order	512
nilpotency class	4
isoclinism	isoclinic class of #13318
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1999
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

999 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:

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

Polycyclic presentation:

2^9|a^2=g,d^2=h,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=dhi,e^b=ei,f^b=fi,g^b=gi,h^b=hi,f^c=fi

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, 54881);