#417910

order	512
nilpotency class	3
isoclinism	isoclinic class of #417909
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15259
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3020	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1899 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)(4,19,22,10)(5,24)(6,13)(7,29)(8,12)(11,26)(14,30,41,58)(15,42)(16,33)(17,32)(18,47)(20,34,46,48)(21,51,53,56)(23,37,28,36)(25,38)(27,39)(31,45)(35,54,49,55)(40,61,63,62)(43,60)(44,59)(50,64,52,57);(1,3)(2,9)(4,20,22,46)(5,15)(6,17)(7,18)(8,16)(10,34,19,48)(11,31)(12,33)(13,32)(14,23,41,28)(21,52,53,50)(24,42)(25,44)(26,45)(27,43)(29,47)(30,36,58,37)(35,62,49,61)(38,59)(39,60)(40,54,63,55)(51,64,56,57);(1,4)(2,10)(3,14)(5,21)(6,28)(7,23)(8,22)(9,30)(11,35)(12,19)(13,37)(15,40)(16,41)(17,46)(18,20)(24,56)(25,55)(26,54)(27,53)(29,36)(31,57)(32,48)(33,58)(34,47)(38,49)(39,51)(42,61)(43,63)(44,50)(45,52)(59,64)(60,62);(1,5,8,27)(2,11,12,38)(3,15,16,43)(4,21,22,53)(6,26,7,25)(9,31,33,59)(10,35,19,49)(13,39,29,24)(14,40,41,63)(17,45,18,44)(20,50,46,52)(23,55,28,54)(30,57,58,64)(32,60,47,42)(34,61,48,62)(36,56,37,51)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306344, #72306450, #72306467
maximal normal subgroup of	#72306344, #72306450, #72306467

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 325106);