#395792

order	512
nilpotency class	3
isoclinism	isoclinic class of #395791
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14727
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1731 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)(3,14,17,33,19,40,18,9)(4,20,24,37,25,36,23,10)(5,26,28,38,29,11,27,39)(15,31,21,51,44,49,43,34)(16,45,47,56,48,41,46,32)(22,50,53,35,55,60,54,57)(42,58,61,59,62,64,52,63);(1,3,8,19)(2,9,13,33)(4,21,25,43)(5,16,29,48)(6,18,7,17)(10,34,37,51)(11,32,26,56)(12,40,30,14)(15,23,44,24)(20,49,36,31)(22,52,55,61)(27,47,28,46)(35,59,57,63)(38,45,39,41)(42,53,62,54)(50,64,60,58);(1,4,6,23,8,25,7,24)(2,10,12,36,13,37,30,20)(3,15,17,43,19,44,18,21)(5,22,27,53,29,55,28,54)(9,31,14,34,33,49,40,51)(11,35,38,60,26,57,39,50)(16,42,46,61,48,62,47,52)(32,58,41,59,56,64,45,63);(1,5)(2,11)(3,16)(4,22)(6,27)(7,28)(8,29)(9,32)(10,35)(12,38)(13,26)(14,41)(15,42)(17,46)(18,47)(19,48)(20,50)(21,52)(23,53)(24,54)(25,55)(30,39)(31,58)(33,56)(34,59)(36,60)(37,57)(40,45)(43,61)(44,62)(49,64)(51,63)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,f^c=fi,h^c=hi,f^d=fi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306450, #72306456, #72306465
maximal normal subgroup of	#72306450, #72306456, #72306465

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 302988);