#483601

order	512
nilpotency class	4
isoclinism	isoclinic class of #33469
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#442
commutator	C₈	C₂³×C₈
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂	C₂×#1580
prime radical	C₂×#508	C₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ C₄×M₃₂ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

463 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=fg,e^2=g,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cg,d^a=di,e^a=eg,g^a=gi,e^b=egi,f^b=fi,g^b=gi,d^c=di,e^c=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224946, #72224947, #72224948, …
maximal normal subgroup of	#72224946, #72224947, #72224948, …
maximal quotient of	#72286654, #72286672, #72319230, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390797);