#461185

order	512
nilpotency class	3
isoclinism	isoclinic class of #455690
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2825
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16382
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

870 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)(3,8,15,29)(4,17)(5,21,22,38)(6,12)(7,11)(9,19)(10,23,35,25)(13,27,40,31)(14,42)(16,30,46,57)(18,49,50,53)(20,37)(24,33)(26,36)(28,48)(32,52,34,51)(39,61)(41,55,47,58)(43,64)(44,59)(45,56)(54,63)(60,62);(1,3)(2,8,11,29)(4,13,19,40)(5,14)(6,16)(7,15)(9,31)(10,28,35,56)(12,30,36,57)(17,27)(18,39,50,43)(20,41,24,47)(21,42,38,59)(22,44)(23,45)(25,48)(26,46)(32,60)(33,58)(34,54)(37,55)(49,61)(51,62,52,63)(53,64);(1,4)(2,9,11,17)(3,13,15,40)(5,18)(6,24)(7,19)(8,27)(10,34,35,32)(12,37,36,33)(14,43,44,39)(16,47,46,41)(20,26)(21,53,38,49)(22,50)(23,52)(25,51)(28,54)(29,31)(30,58)(42,64)(45,62,48,63)(55,57)(56,60)(59,61);(1,5)(2,10)(3,14)(4,18)(6,25)(7,22)(8,28)(9,32)(11,35)(12,38)(13,39)(15,44)(16,48)(17,34)(19,50)(20,52)(21,36)(23,26)(24,51)(27,54)(29,56)(30,59)(31,60)(33,49)(37,53)(40,43)(41,63)(42,57)(45,46)(47,62)(55,64)(58,61)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 368381);