#417908

order	512
nilpotency class	3
isoclinism	isoclinic class of #417907
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×#585	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1910 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)(4,19,22,38)(5,24)(6,12)(7,29)(8,13)(10,23,37,28)(11,25)(14,41,43,35)(15,45)(16,32)(17,33)(18,34)(20,30,44,49)(21,52,54,36)(26,40)(27,39)(31,47)(42,57,63,64)(46,58)(48,59)(50,55,62,56)(51,60,53,61);(1,3)(2,9)(4,20,22,44)(5,15)(6,17)(7,16)(8,18)(10,35,37,41)(11,31)(12,33)(13,34)(14,23,43,28)(19,49,38,30)(21,53,54,51)(24,45)(25,47)(26,46)(27,48)(29,32)(36,61,52,60)(39,59)(40,58)(42,56,63,55)(50,57,62,64);(1,4)(2,10)(3,14)(5,21)(6,28)(7,23)(8,22)(9,30)(11,36)(12,38)(13,37)(15,42)(16,20)(17,44)(18,43)(19,29)(24,50)(25,56)(26,55)(27,54)(31,57)(32,35)(33,41)(34,49)(39,62)(40,52)(45,60)(46,51)(47,53)(48,63)(58,64)(59,61);(1,5,8,27)(2,11,13,40)(3,15,18,48)(4,21,22,54)(6,25,7,26)(9,31,34,58)(10,36,37,52)(12,39,29,24)(14,42,43,63)(16,46,17,47)(19,50,38,62)(20,51,44,53)(23,55,28,56)(30,57,49,64)(32,45,33,59)(35,60,41,61)

Polycyclic presentation:

2^9|a^2=e,b^2=e,d^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,f^b=fi,e^c=ei,f^c=fi,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	#72264934, #72265041
maximal normal subgroup of	#72264934, #72265041
maximal quotient of	#72314035

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 325104);