#147485

order	512
nilpotency class	4
isoclinism	isoclinic class of #13030
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1879
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10931
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

713 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54681);