#426022

order	512
nilpotency class	3
isoclinism	isoclinic class of #417630
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2669
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15345
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

966 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)(3,9)(4,20,23,37)(5,25)(6,12)(7,13)(8,14)(10,24,36,29)(11,26)(15,34,42,57)(16,44)(17,32)(18,40)(19,33)(21,30,43,48)(22,51,53,35)(27,39)(28,38)(31,45)(41,63,64,56)(46,58)(47,59)(49,54,62,55)(50,60,52,61);(1,3,6,17,8,19,7,18)(2,9,12,32,14,33,13,40)(4,21,29,42,23,43,24,15)(5,16,26,45,28,46,27,47)(10,34,37,30,36,57,20,48)(11,31,38,58,39,59,25,44)(22,52,55,41,53,50,54,64)(35,61,62,63,51,60,49,56);(1,4,6,29,8,23,7,24)(2,10,12,37,14,36,13,20)(3,15,17,21,19,42,18,43)(5,22,26,55,28,53,27,54)(9,30,32,57,33,48,40,34)(11,35,38,62,39,51,25,49)(16,41,45,50,46,64,47,52)(31,56,58,61,59,63,44,60);(1,5)(2,11)(3,16)(4,22)(6,26)(7,27)(8,28)(9,31)(10,35)(12,38)(13,25)(14,39)(15,41)(17,45)(18,47)(19,46)(20,49)(21,50)(23,53)(24,54)(29,55)(30,56)(32,58)(33,59)(34,60)(36,51)(37,62)(40,44)(42,64)(43,52)(48,63)(57,61)

Polycyclic presentation:

2^9|a^2=e,b^2=ef,c^2=ef,f^2=i,g^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,g^b=gi,e^c=ei,f^c=fi,g^c=gi,g^d=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, 333218);