#425093

order	512
nilpotency class	3
isoclinism	isoclinic class of #418871
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2697
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15328
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1003 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:

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

Polycyclic presentation:

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

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, 332289);