#449877

order	512
nilpotency class	3
isoclinism	isoclinic class of #428398
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2609
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15952
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1003 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)(3,8,15,27)(4,9,19,31)(5,10)(6,12)(7,11)(13,25)(14,38,21,26)(16,28,40,55)(17,29)(18,43,46,30)(20,32,47,51)(22,33)(23,34)(24,35)(36,52)(37,53)(39,61,49,54)(41,56)(42,57)(44,62)(45,63,64,58)(48,60)(50,59);(1,3)(2,8,11,27)(4,17,19,42)(5,21,22,14)(6,16)(7,15)(9,29)(10,26)(12,28,35,55)(13,20,37,47)(18,44)(23,49,50,39)(24,40)(25,51)(30,56,43,62)(31,57)(32,53)(33,38)(34,54)(36,45)(41,46)(48,64)(52,58,60,63)(59,61);(1,4)(2,9,11,31)(3,13,15,37)(5,18)(6,20)(7,19)(8,25)(10,30,33,43)(12,32,35,51)(14,36,21,48)(16,17,40,42)(22,46)(23,45)(24,47)(26,52)(27,53)(28,29)(34,58,59,63)(38,60)(39,41,49,44)(50,64)(54,56)(55,57)(61,62);(1,5)(2,10)(3,14)(4,18)(6,23)(7,22)(8,26)(9,30)(11,33)(12,34)(13,36)(15,21)(16,39)(17,41)(19,46)(20,45)(24,50)(25,52)(27,38)(28,54)(29,56)(31,43)(32,58)(35,59)(37,48)(40,49)(42,44)(47,64)(51,63)(53,60)(55,61)(57,62)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 357073);