#364607

order	512
nilpotency class	3
isoclinism	isoclinic class of #364604
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#13967
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3020	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2300 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)(3,15,18,9,19,33,30,34)(4,20,23,10,25,37,24,36)(5,26,27,11,29,39,28,38)(16,42,45,49,21,31,44,40)(17,32,47,57,48,58,46,41)(22,35,53,55,54,59,52,50)(43,60,62,61,51,64,63,56);(1,3,7,18,8,19,6,30)(2,9,12,33,14,34,13,15)(4,21,23,44,25,16,24,45)(5,17,27,47,29,48,28,46)(10,31,37,40,36,42,20,49)(11,32,39,57,38,58,26,41)(22,51,53,63,54,43,52,62)(35,56,55,60,59,61,50,64);(1,4)(2,10)(3,16)(5,22)(6,23)(7,24)(8,25)(9,31)(11,35)(12,20)(13,37)(14,36)(15,40)(17,43)(18,44)(19,21)(26,55)(27,52)(28,53)(29,54)(30,45)(32,56)(33,49)(34,42)(38,59)(39,50)(41,60)(46,62)(47,63)(48,51)(57,64)(58,61);(1,5,8,29)(2,11,14,38)(3,17,19,48)(4,22,25,54)(6,28,7,27)(9,32,34,58)(10,35,36,59)(12,39,13,26)(15,41,33,57)(16,43,21,51)(18,47,30,46)(20,50,37,55)(23,53,24,52)(31,56,42,61)(40,60,49,64)(44,63,45,62)

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=i,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,e^b=ei,f^b=fi,g^b=gi,f^c=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, 271803);