#496951

order	512
nilpotency class	3
isoclinism	isoclinic class of #23994
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#535
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#22135
prime radical	#3067	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

516 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=e,e^2=i,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=ceh,f^b=fi,f^c=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320560, #72320718, #72320755
maximal normal subgroup of	#72320560, #72320718, #72320755

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404147);