#430065

order	512
nilpotency class	3
isoclinism	isoclinic class of #430064
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15412
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23434	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1476 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)(3,9,16,31)(4,10)(5,23)(6,13)(7,28)(8,12)(11,26)(14,41,43,29)(15,44,45,57)(17,32,18,49)(19,50,48,33)(20,52)(21,37)(22,35)(24,40)(25,39)(27,36)(30,46,38,47)(34,54)(42,63,64,59)(51,58,53,62)(55,61)(56,60);(1,3)(2,9,12,31)(4,19,21,48)(5,15,24,45)(6,17)(7,18)(8,16)(10,33)(11,38)(13,32,28,49)(14,22,43,27)(20,53)(23,57)(25,46,26,47)(29,36)(30,39)(34,59,60,63)(35,41)(37,50)(40,44)(42,56)(51,55)(52,62,61,58)(54,64);(1,4,8,21)(2,10,12,37)(3,14,16,43)(5,20,24,55)(6,27,7,22)(9,29,31,41)(11,34,39,60)(13,36,28,35)(15,42,45,64)(17,48,18,19)(23,52,40,61)(25,56,26,54)(30,58,38,62)(32,50,49,33)(44,59,57,63)(46,51,47,53);(1,5)(2,11)(3,15,16,45)(4,20)(6,26)(7,25)(8,24)(9,30,31,38)(10,34)(12,39)(13,40)(14,42,43,64)(17,47,18,46)(19,51,48,53)(21,55)(22,56)(23,28)(27,54)(29,58,41,62)(32,57,49,44)(33,59,50,63)(35,52)(36,61)(37,60)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72264934, #72264954, #72265017
maximal normal subgroup of	#72264934, #72264954, #72265017
maximal quotient of	#72314033

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 337261);