#461178

order	512
nilpotency class	3
isoclinism	isoclinic class of #461177
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16382
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

863 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)(3,13)(4,18,21,9)(5,23,24,35)(6,12)(7,11)(8,16)(10,25,33,26)(14,36,19,28)(15,41,43,55)(17,38)(20,50)(22,48,53,32)(27,34)(29,46,37,44)(30,45)(31,51)(39,61)(40,60,49,58)(42,64)(47,54)(52,56)(57,62)(59,63);(1,3)(2,8,11,13)(4,19)(5,15)(6,17)(7,16)(9,28,18,36)(10,29,33,37)(12,30,34,38)(14,21)(20,42)(22,49)(23,55,35,41)(24,43)(25,44)(26,46)(27,45)(31,57,47,59)(32,58,48,60)(39,52)(40,53)(50,61,56,64)(51,63)(54,62);(1,4)(2,9)(3,14,16,19)(5,20)(6,22)(7,21)(8,28,13,36)(10,31)(11,18)(12,32)(15,42,43,39)(17,40,45,49)(23,56)(24,52)(25,51)(26,54)(27,53)(29,59,37,57)(30,58,38,60)(33,47)(34,48)(35,50)(41,64,55,61)(44,63,46,62);(1,5,7,24)(2,10,11,33)(3,15,16,43)(4,20,21,52)(6,26,27,25)(8,29,13,37)(9,31,18,47)(12,35,34,23)(14,39,19,42)(17,46,45,44)(22,54,53,51)(28,57,36,59)(30,41,38,55)(32,50,48,56)(40,63,49,62)(58,61,60,64)

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286066, #72286106, #72286267
maximal normal subgroup of	#72286066, #72286106, #72286267

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 368374);