#461423

order	512
nilpotency class	3
isoclinism	isoclinic class of #457608
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2735
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16391
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1416 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)(3,14)(4,20)(5,25,28,39)(6,30)(7,12)(8,13)(9,17)(10,22)(11,29,38,27)(15,31)(16,32,49,61)(18,34)(19,33)(21,35,53,55)(23,36)(24,52)(26,51,54,37)(40,44)(41,48,60,47)(42,59,63,62)(43,50)(45,58)(46,57,56,64);(1,3)(2,9)(4,15)(5,16)(6,18)(7,17)(8,19)(10,31)(11,32)(12,33)(13,34)(14,30)(20,50)(21,42)(22,44)(23,43)(24,45)(25,41)(26,56)(27,48)(28,49)(29,47)(35,57)(36,58)(37,62)(38,61)(39,60)(40,52)(46,54)(51,59)(53,63)(55,64);(1,4,6,22,8,24,7,23)(2,10,12,20,13,36,30,52)(3,15,18,44,19,45,17,43)(5,26,27,53,28,54,29,21)(9,31,33,50,34,58,14,40)(11,37,25,55,38,51,39,35)(16,46,48,42,49,56,47,63)(32,59,41,57,61,62,60,64);(1,5)(2,11)(3,16)(4,21)(6,29)(7,27)(8,28)(9,32)(10,35)(12,39)(13,38)(14,41)(15,42)(17,48)(18,47)(19,49)(20,51)(22,54)(23,26)(24,53)(25,30)(31,57)(33,60)(34,61)(36,55)(37,52)(40,62)(43,56)(44,46)(45,63)(50,59)(58,64)

Polycyclic presentation:

2^9|a^2=g,c^2=f,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,e^b=ei,h^b=hi,d^c=de,e^d=ei,f^d=fi,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, 368619);