#434422

order	512
nilpotency class	3
isoclinism	isoclinic class of #420517
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15501
prime radical	#23412	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1137 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)(3,14)(4,20,24,35)(5,25)(6,12)(7,13)(8,17)(9,23,36,54)(10,28)(11,29)(15,33,21,30)(16,37)(18,32)(19,40)(22,50,53,48)(26,31)(27,38)(34,62,56,52)(39,58)(41,44,47,43)(42,61,49,59)(45,60)(46,55)(51,63,57,64);(1,3)(2,8)(4,21)(5,26)(6,18)(7,19)(9,33)(10,37)(11,40)(12,14)(13,32)(15,24)(16,27)(17,29)(20,47)(22,51)(23,43)(25,55)(28,45)(30,36)(31,38)(34,63)(35,41)(39,60)(42,62)(44,54)(46,58)(48,59)(49,52)(50,61)(53,57)(56,64);(1,4,7,24)(2,9)(3,15)(5,22,28,53)(6,23,29,54)(8,30,32,33)(10,34)(11,35)(12,20)(13,36)(14,41,40,47)(16,42)(17,43)(18,44)(19,21)(25,56)(26,57)(27,52,58,62)(31,59,60,61)(37,64,55,63)(38,50)(39,48)(45,51)(46,49);(1,5,6,27,7,28,29,58)(2,10,11,38,13,25,12,39)(3,16,17,26,19,46,18,45)(4,22,23,52,24,53,54,62)(8,31,14,37,32,60,40,55)(9,34,35,50,36,56,20,48)(15,42,43,57,21,49,44,51)(30,59,41,64,33,61,47,63)

Polycyclic presentation:

2^9|a^2=g,c^2=ef,d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,e^b=ei,h^b=hi,g^c=gi,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, 341618);