#502408

order	512
nilpotency class	4
isoclinism	isoclinic class of #501994
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3341
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#25339
prime radical	#28249	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#443 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

500 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:

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

Polycyclic presentation:

2^9|a^2=efh,b^2=f,d^2=e,e^2=i,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,f^c=fh,g^c=gh,h^c=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 409604);