#421700

order	512
nilpotency class	3
isoclinism	isoclinic class of #418530
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2676
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15296
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

805 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=ef,c^2=e,d^2=e,e^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,e^b=ei,f^b=fi,g^b=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, 328896);