#412076

order	512
nilpotency class	3
isoclinism	isoclinic class of #400512
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2491
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15126
prime radical	C₂×#3067	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1303 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)(3,15)(4,20)(5,26)(6,14)(7,12)(8,13)(9,18)(10,24)(11,28)(16,40)(17,44)(19,39)(21,49)(22,52)(23,35)(25,36)(27,38)(29,37)(30,33)(31,43)(32,46)(34,53)(41,45)(42,61)(47,51)(48,55)(50,58)(54,56)(57,59)(60,62)(63,64);(1,3)(2,9)(4,21)(5,17)(6,30)(7,18)(8,19)(10,31)(11,32)(12,39)(13,33)(14,15)(16,24)(20,47)(22,50)(23,43)(25,51)(26,41)(27,57)(28,45)(29,46)(34,58)(35,49)(36,40)(37,59)(38,44)(42,54)(48,61)(52,63)(53,62)(55,64)(56,60);(1,4,6,23,8,24,7,25)(2,10,12,35,13,20,14,36)(3,16,18,43,19,21,30,51)(5,22,27,53,29,54,28,55)(9,31,15,40,33,47,39,49)(11,34,37,52,38,48,26,56)(17,42,45,62,46,50,57,64)(32,58,41,60,44,61,59,63);(1,5,7,28,8,29,6,27)(2,11,14,26,13,38,12,37)(3,17,18,45,19,46,30,57)(4,22,25,55,24,54,23,53)(9,32,15,41,33,44,39,59)(10,34,36,56,20,48,35,52)(16,42,43,62,21,50,51,64)(31,58,40,60,47,61,49,63)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 319272);