#437721

order	512
nilpotency class	3
isoclinism	isoclinic class of #427841
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2778
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15610
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

886 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)(3,8,15,31)(4,17,20,9)(5,22)(6,25)(7,11)(10,24)(12,28)(13,33)(14,42,43,55)(16,46,45,32)(18,29)(19,51,52,60)(21,35,26,38)(23,39)(27,37)(30,48,36,44)(34,53,49,54)(40,61)(41,57)(47,58)(50,64)(56,63)(59,62);(1,3)(2,8,11,31)(4,18)(5,14,23,43)(6,16)(7,15)(9,33,17,29)(10,36)(12,32,25,46)(13,20)(19,50,52,40)(21,47)(22,55)(24,44,27,48)(26,41)(28,45)(30,37)(34,56)(35,58,38,57)(39,42)(49,59)(51,61)(53,63,54,62)(60,64);(1,4)(2,9)(3,13,15,18)(5,19)(6,26)(7,20)(8,29,31,33)(10,34)(11,17)(12,38)(14,40,43,50)(16,47,45,41)(21,28)(22,51)(23,52)(24,54)(25,35)(27,53)(30,56,36,59)(32,58,46,57)(37,49)(39,60)(42,61,55,64)(44,63,48,62);(1,5)(2,10)(3,14,15,43)(4,19)(6,27)(7,23)(8,30,31,36)(9,34)(11,37)(12,39)(13,40,18,50)(16,48,45,44)(17,49)(20,52)(21,54)(22,25)(24,28)(26,53)(29,56,33,59)(32,55,46,42)(35,51)(38,60)(41,63,47,62)(57,61,58,64)

Polycyclic presentation:

2^9|a^2=efg,b^2=f,c^2=ef,d^2=ef,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^c=gi,h^c=hi,e^d=ei,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, 344917);