#494901

order	512
nilpotency class	4
isoclinism	isoclinic class of #33618
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#466
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#1934
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

767 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=e,e^2=hi,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,h^a=hi,d^b=di,f^c=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402097);