#434723

order	512
nilpotency class	3
isoclinism	isoclinic class of #434722
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15506
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1028 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 341919);