#436635

order	512
nilpotency class	3
isoclinism	isoclinic class of #419463
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2557
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15566
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

924 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,d^2=ef,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,g^b=gi,g^c=gi,h^c=hi,e^d=ei,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, 343831);