#500686

order	512
nilpotency class	4
isoclinism	isoclinic class of #500685
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24208
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1134 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)(3,17)(4,24)(5,14)(6,15)(7,12)(8,13)(9,16)(10,20)(11,26)(18,47)(19,46)(21,35)(22,37)(23,36)(25,56)(27,59)(28,43)(29,55)(30,42)(31,41)(32,44)(33,52)(34,51)(38,60)(39,57)(40,58)(45,48)(49,54)(50,62)(53,63)(61,64);(1,3,6,20,9,23,8,22)(2,10,13,17,16,37,15,36)(4,25,27,49,29,53,28,18)(5,19,7,21,31,51,30,50)(11,38,39,33,40,45,44,61)(12,34,14,35,42,46,41,62)(24,54,43,56,55,47,59,63)(26,52,32,60,58,64,57,48);(1,4)(2,11)(3,18)(5,26)(6,27)(7,32)(8,28)(9,29)(10,33)(12,24)(13,39)(14,43)(15,44)(16,40)(17,45)(19,48)(20,25)(21,52)(22,53)(23,49)(30,57)(31,58)(34,56)(35,47)(36,38)(37,61)(41,59)(42,55)(46,63)(50,64)(51,60)(54,62);(1,5,8,30,9,31,6,7)(2,12,15,41,16,42,13,14)(3,19,22,50,23,51,20,21)(4,26,28,57,29,58,27,32)(10,34,36,62,37,46,17,35)(11,24,44,59,40,55,39,43)(18,48,53,64,49,60,25,52)(33,56,38,54,61,63,45,47)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283860
maximal normal subgroup of	#72283860

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407882);