#453832

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	#16143
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:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=fh,c^2=e,d^2=ef,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^c=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, 361028);