#368560

order	512
nilpotency class	3
isoclinism	isoclinic class of #368558
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14020
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1143 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284899, #72286194, #72286206
maximal normal subgroup of	#72284899, #72286194, #72286206

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 275756);