#422904

order	512
nilpotency class	3
isoclinism	isoclinic class of #422903
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15310
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1858 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)(5,26)(6,13)(7,12)(8,14)(9,18)(11,29)(16,35)(17,48)(19,34)(20,33)(21,51)(22,53)(23,42)(24,37)(25,38)(27,39)(28,41)(30,40)(31,46)(32,49)(36,55)(43,47)(44,50)(45,59)(52,60)(54,64)(56,61)(57,62)(58,63);(1,3,6,18,8,19,7,20)(2,9,12,15,14,33,13,34)(4,21,23,16,25,46,24,47)(5,27,28,49,30,50,29,17)(10,35,37,51,38,43,42,31)(11,39,40,48,41,44,26,32)(22,54,55,58,57,45,56,52)(36,59,53,63,61,64,62,60);(1,4)(2,10)(3,16,19,47)(5,22)(6,24)(7,23)(8,25)(9,31,33,51)(11,36)(12,42)(13,37)(14,38)(15,43,34,35)(17,45,49,54)(18,21,20,46)(26,53)(27,58,50,52)(28,56)(29,55)(30,57)(32,59,48,64)(39,60,44,63)(40,62)(41,61);(1,5,8,30)(2,11,14,41)(3,17,19,49)(4,22,25,57)(6,29,7,28)(9,32,33,48)(10,36,38,61)(12,26,13,40)(15,44,34,39)(16,45,47,54)(18,50,20,27)(21,52,46,58)(23,56,24,55)(31,59,51,64)(35,60,43,63)(37,62,42,53)

Polycyclic presentation:

2^9|b^2=e,c^2=f,d^2=i,e^2=i,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,f^b=fi,e^c=ei,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	#72306392, #72306455, #72306459
maximal normal subgroup of	#72306392, #72306455, #72306459

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 330100);