#500649

order	512
nilpotency class	4
isoclinism	isoclinic class of #500646
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24196
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3414	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2960 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

744 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)(3,9)(4,19)(5,11)(6,29)(7,35)(8,15)(10,23)(12,33)(13,49)(14,32)(16,54)(17,47)(18,28)(20,38)(21,59)(22,43)(24,51)(25,58)(26,63)(27,30)(31,53)(34,52)(36,39)(37,48)(40,45)(41,62)(42,55)(44,64)(46,50)(56,60)(57,61);(1,3,8,11)(2,9,15,5)(4,20,22,58)(6,30,34,63)(7,36,32,18)(10,40,42,54)(12,47,13,50)(14,28,35,39)(16,55,45,23)(17,49,46,33)(19,25,43,38)(21,51,60,48)(24,59,37,56)(26,29,27,52)(31,57,64,41)(44,61,53,62);(1,4)(2,10)(3,16)(5,25)(6,31)(7,37)(8,22)(9,38)(11,45)(12,48)(13,51)(14,53)(15,42)(17,56)(18,57)(19,29)(20,30)(21,28)(23,33)(24,32)(26,62)(27,61)(34,64)(35,44)(36,41)(39,60)(40,50)(43,52)(46,59)(47,54)(49,55)(58,63);(1,5,8,9)(2,11,15,3)(4,21)(6,26,34,27)(7,28,32,39)(10,41)(12,46,13,17)(14,36,35,18)(16,53)(19,56)(20,51)(22,60)(23,61)(24,25)(29,63,52,30)(31,40)(33,50,49,47)(37,38)(42,57)(43,59)(44,45)(48,58)(54,64)(55,62)

Polycyclic presentation:

2^9|b^2=i,d^2=ei,f^2=hi,g^2=hi,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=gh,h^c=hi,g^d=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307725
maximal normal subgroup of	#72307725

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407845);