#464075

order	512
nilpotency class	3
isoclinism	isoclinic class of #464074
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16500
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23412	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1413 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)(4,10)(5,23)(6,14)(7,12)(8,13)(9,18)(11,27)(16,42)(17,31)(19,41)(20,37)(21,40)(22,36)(24,34)(25,39)(26,38)(28,35)(29,32)(30,46)(33,45)(43,49)(44,55)(47,56)(48,62)(50,59)(51,58)(52,61)(53,54)(57,64)(60,63);(1,3)(2,9,13,32)(4,16)(5,17)(6,18)(7,29)(8,19)(10,33,36,30)(11,31,39,59)(12,15,14,41)(20,44)(21,45)(22,47)(23,43,38,58)(24,48)(25,49)(26,50)(27,51)(28,46)(34,60,54,55)(35,42,40,56)(37,62,61,57)(52,63)(53,64);(1,4,8,22)(2,10,13,36)(3,16,19,47)(5,24,26,53)(6,28,7,21)(9,30,32,33)(11,37,39,61)(12,40,14,35)(15,42,41,56)(17,48,50,64)(18,46,29,45)(20,27,52,25)(23,54,38,34)(31,57,59,62)(43,60,58,55)(44,51,63,49);(1,5)(2,11)(3,17)(4,20)(6,27)(7,25)(8,26)(9,31)(10,34)(12,23)(13,39)(14,38)(15,43)(16,44)(18,51)(19,50)(21,53)(22,52)(24,28)(29,49)(30,55)(32,59)(33,60)(35,61)(36,54)(37,40)(41,58)(42,57)(45,64)(46,48)(47,63)(56,62)

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72264914, #72264949, #72265041
maximal normal subgroup of	#72264914, #72264949, #72265041
maximal quotient of	#72265536

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 371271);