#364601

order	512
nilpotency class	3
isoclinism	isoclinic class of #364600
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×#585	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2319 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)(3,15)(4,20,25,40)(5,26)(6,13)(7,12)(8,14)(9,18)(10,24,37,23)(11,27)(16,43,46,31)(17,32)(19,33)(21,35,45,41)(22,36,54,60)(28,38)(29,39)(30,34)(42,47)(44,61,63,59)(48,57)(49,58)(50,53,55,52)(51,62,64,56);(1,3)(2,9)(4,21,25,45)(5,17)(6,30)(7,18)(8,19)(10,35,37,41)(11,32)(12,33)(13,15)(14,34)(16,23,46,24)(20,43,40,31)(22,51,54,64)(26,42)(27,48)(28,47)(29,49)(36,59,60,61)(38,58)(39,57)(44,53,63,52)(50,62,55,56);(1,4)(2,10)(3,16)(5,22)(6,23)(7,24)(8,25)(9,31)(11,36)(12,20)(13,40)(14,37)(15,41)(17,44)(18,45)(19,46)(21,30)(26,55)(27,52)(28,53)(29,54)(32,56)(33,35)(34,43)(38,60)(39,50)(42,61)(47,51)(48,64)(49,63)(57,59)(58,62);(1,5,8,29)(2,11,14,38)(3,17,19,49)(4,22,25,54)(6,28,7,27)(9,32,34,58)(10,36,37,60)(12,39,13,26)(15,42,33,57)(16,44,46,63)(18,48,30,47)(20,50,40,55)(21,51,45,64)(23,53,24,52)(31,56,43,62)(35,59,41,61)

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=i,e^2=i,f^2=i,g^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,f^c=fi,g^c=gi,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, 271797);