#376204

order	512
nilpotency class	3
isoclinism	isoclinic class of #376201
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

1472 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)(3,15,20,34,19,33,18,9)(4,21,25,10,26,36,24,37)(5,27,29,38,30,39,28,11)(16,42,45,40,22,31,44,56)(17,46,48,57,49,32,47,41)(23,35,53,58,54,59,52,50)(43,60,63,55,51,64,62,61);(1,3,6,18,8,19,7,20)(2,9,12,15,14,34,13,33)(4,22,24,45,26,16,25,44)(5,17,28,47,30,49,29,48)(10,31,36,56,37,42,21,40)(11,32,27,41,38,46,39,57)(23,51,52,63,54,43,53,62)(35,55,58,64,59,61,50,60);(1,4,8,26)(2,10,14,37)(3,16,19,22)(5,23,30,54)(6,25,7,24)(9,31,34,42)(11,35,38,59)(12,21,13,36)(15,40,33,56)(17,43,49,51)(18,45,20,44)(27,50,39,58)(28,53,29,52)(32,55,46,61)(41,60,57,64)(47,63,48,62);(1,5)(2,11,14,38)(3,17)(4,23)(6,28)(7,29)(8,30)(9,32,34,46)(10,35,37,59)(12,27,13,39)(15,41,33,57)(16,43)(18,47)(19,49)(20,48)(21,50,36,58)(22,51)(24,52)(25,53)(26,54)(31,55,42,61)(40,60,56,64)(44,62)(45,63)

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=i,d^2=eh,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,g^b=gi,e^c=ei,f^c=fi,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, 283400);