#500746

order	512
nilpotency class	4
isoclinism	isoclinic class of #500743
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24248
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2929 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1011 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307114, #72307124, C₂×#500746
maximal normal subgroup of	#72307114, #72307124, C₂×#500746
maximal quotient of	C₂×#500746

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407942);