#423836

order	512
nilpotency class	3
isoclinism	isoclinic class of #418470
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2676
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15316
prime radical	#60469	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1271 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:

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

Polycyclic presentation:

2^9|a^2=e,c^2=f,d^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,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, 331032);