#368416

order	512
nilpotency class	3
isoclinism	isoclinic class of #24295
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2408
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14018
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1281 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=g,d^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,g^b=gi,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, 275612);