#453881

order	512
nilpotency class	3
isoclinism	isoclinic class of #427666
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2527
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16194
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

531 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)(3,8,16,32)(4,18)(5,23)(6,26)(7,11)(9,22)(10,24)(12,13)(14,40,19,49)(15,35,44,30)(17,45,39,31)(20,48)(21,37)(25,36)(27,34)(28,38)(29,46,56,42)(33,53)(41,61,50,64)(43,57,47,58)(51,54)(52,59)(55,62,60,63);(1,3,13,39,7,16,6,17)(2,8,26,45,11,32,12,31)(4,19,22,46,21,14,27,42)(5,15,28,47,24,44,25,43)(9,29,18,40,34,56,37,49)(10,35,38,57,23,30,36,58)(20,41,51,62,52,50,53,63)(33,55,59,64,54,60,48,61);(1,4)(2,9)(3,14,16,19)(5,20)(6,27)(7,21)(8,29,32,56)(10,33)(11,34)(12,37)(13,22)(15,41,44,50)(17,46,39,42)(18,26)(23,54)(24,52)(25,53)(28,51)(30,55,35,60)(31,49,45,40)(36,48)(38,59)(43,63,47,62)(57,61,58,64);(1,5,6,28,7,24,13,25)(2,10,26,36,11,23,12,38)(3,15,39,43,16,44,17,47)(4,20,27,51,21,52,22,53)(8,30,31,58,32,35,45,57)(9,33,18,48,34,54,37,59)(14,41,42,63,19,50,46,62)(29,55,49,64,56,60,40,61)

Polycyclic presentation:

2^9|a^2=eg,b^2=fh,c^2=eg,d^2=ef,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=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, 361077);