#431132

order	512
nilpotency class	3
isoclinism	isoclinic class of #420055
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2476
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15431
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

802 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)(5,11,29,42)(6,13)(7,12)(8,14)(9,18)(10,25)(16,31)(17,43,48,39)(19,34)(20,33)(22,35)(23,49,54,61)(24,38)(26,37)(27,55,47,32)(28,41,30,40)(36,53,60,52)(44,56,51,64)(45,57)(46,58)(50,59,63,62);(1,3,6,18,8,20,7,19)(2,9,12,15,14,34,13,33)(4,22,24,16,26,45,25,46)(5,27,28,48,29,47,30,17)(10,35,37,58,38,57,21,31)(11,39,40,55,42,43,41,32)(23,51,52,63,54,44,53,50)(36,56,49,62,60,64,61,59);(1,4,7,25,8,26,6,24)(2,10,13,21,14,38,12,37)(3,16,19,22,20,46,18,45)(5,23,28,52,29,54,30,53)(9,31,33,57,34,58,15,35)(11,36,40,49,42,60,41,61)(17,44,27,50,48,51,47,63)(32,56,39,62,55,64,43,59);(1,5,7,28,8,29,6,30)(2,11,13,40,14,42,12,41)(3,17,19,27,20,48,18,47)(4,23,25,52,26,54,24,53)(9,32,33,39,34,55,15,43)(10,36,21,49,38,60,37,61)(16,44,22,50,46,51,45,63)(31,56,57,62,58,64,35,59)

Polycyclic presentation:

2^9|a^2=ef,b^2=e,c^2=f,d^2=f,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=de,g^b=gi,g^c=gi,e^d=ei,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, 338328);