#454669

order	512
nilpotency class	3
isoclinism	isoclinic class of #454668
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16255
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60368	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1746 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318867, #72318935, #72319003
maximal normal subgroup of	#72318867, #72318935, #72319003

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 361865);