#450711

order	512
nilpotency class	3
isoclinism	isoclinic class of #417617
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2678
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15975
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

817 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=ef,c^2=ef,d^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,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, 357907);