#433538

order	512
nilpotency class	3
isoclinism	isoclinic class of #425083
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2576
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15484
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

747 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)(3,13)(4,18,21,9)(5,23,25,39)(6,12)(7,11)(8,16)(10,26,36,28)(14,40,19,30)(15,44,24,55)(17,41)(20,51)(22,38,27,34)(29,37)(31,48,35,45)(32,46)(33,53)(42,61)(43,58,47,57)(49,54)(50,64)(52,60)(56,62)(59,63);(1,3)(2,8,11,13)(4,19)(5,24,25,15)(6,17)(7,16)(9,30,18,40)(10,35)(12,32,37,41)(14,21)(20,42,52,50)(22,47)(23,44)(26,48,28,45)(27,43)(29,46)(31,36)(33,59)(34,57,38,58)(39,55)(49,56)(51,64)(53,62,54,63)(60,61);(1,4)(2,9)(3,14,16,19)(5,20)(6,27)(7,21)(8,30,13,40)(10,33)(11,18)(12,38)(15,42,24,50)(17,47,46,43)(22,29)(23,51)(25,52)(26,54)(28,53)(31,56,35,59)(32,58,41,57)(34,37)(36,49)(39,60)(44,64,55,61)(45,63,48,62);(1,5)(2,10)(3,15)(4,20)(6,28)(7,25)(8,31)(9,33)(11,36)(12,39)(13,35)(14,42)(16,24)(17,48)(18,49)(19,50)(21,52)(22,54)(23,37)(26,29)(27,53)(30,56)(32,44)(34,51)(38,60)(40,59)(41,55)(43,63)(45,46)(47,62)(57,61)(58,64)

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,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^c=gi,h^c=hi,f^d=fi,h^d=hi

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, 340734);