#368890

order	512
nilpotency class	3
isoclinism	isoclinic class of #368888
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14023
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1623 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 276086);