#376208

order	512
nilpotency class	3
isoclinism	isoclinic class of #376205
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14175
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

1480 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=i,d^2=eh,e^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,g^b=gi,e^c=ei,h^c=hi,f^d=fi

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, 283404);