#478827

order	512
nilpotency class	3
isoclinism	isoclinic class of #478826
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#17170
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

635 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)(4,21)(5,27,32,44)(6,13)(7,12)(8,14)(9,18)(10,25)(11,31,42,30)(16,37)(17,49,52,34)(19,36)(20,35)(22,33)(23,54,55,41)(24,39)(26,43)(28,40,51,61)(29,53,56,38)(45,47)(46,63,64,58)(48,59)(50,60,57,62);(1,3,8,20)(2,9,14,36)(4,22,25,48)(5,28,32,51)(6,19,7,18)(10,37,21,45)(11,40,42,61)(12,35,13,15)(16,26,47,24)(17,31,52,30)(23,46,55,64)(27,34,44,49)(29,57,56,50)(33,43,59,39)(38,58,53,63)(41,62,54,60);(1,4,6,26,8,25,7,24)(2,10,12,43,14,21,13,39)(3,16,18,48,20,47,19,22)(5,29,30,23,32,56,31,55)(9,33,15,45,36,59,35,37)(11,41,27,38,42,54,44,53)(17,50,28,46,52,57,51,64)(34,60,40,58,49,62,61,63);(1,5,6,30,8,32,7,31)(2,11,12,27,14,42,13,44)(3,17,18,28,20,52,19,51)(4,23,24,29,25,55,26,56)(9,34,15,40,36,49,35,61)(10,38,39,41,21,53,43,54)(16,46,22,50,47,64,48,57)(33,58,37,60,59,63,45,62)

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 386023);