#499880

order	512
nilpotency class	4
isoclinism	isoclinic class of #499879
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#23900
commutator, Frattini, cosocle	C₂×#41	C₂⁴
prime radical	#60390	C₂
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×2₋¹⁺⁴ ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

1814 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)(3,14)(4,9)(5,10)(6,27)(7,13)(8,16)(11,25)(12,30)(15,46)(17,40)(18,51)(19,53)(20,35)(21,43)(22,38)(23,33)(24,36)(26,41)(28,45)(29,37)(31,49)(32,57)(34,54)(39,59)(42,50)(44,48)(47,60)(52,63)(55,61)(56,64)(58,62);(1,3)(2,8)(4,19)(5,23)(6,28)(7,17)(9,34)(10,14)(11,40)(12,42)(13,33)(15,21)(16,25)(18,30)(20,55)(22,52)(24,54)(26,50)(27,32)(29,48)(31,37)(35,53)(36,63)(38,61)(39,51)(41,45)(43,60)(44,62)(46,64)(47,58)(49,56)(57,59);(1,4,7,22)(2,9,13,38)(3,15,17,48)(5,24,25,20)(6,29,30,21)(8,31,33,60)(10,35,11,36)(12,43,27,37)(14,44,40,46)(16,49,23,47)(18,52,28,19)(26,56,59,58)(32,61,42,34)(39,64,41,62)(45,63,51,53)(50,54,57,55);(2,10)(5,25)(8,14)(9,35)(11,13)(12,39)(16,23)(20,24)(26,59)(27,41)(31,44)(32,45)(33,40)(34,53)(36,38)(37,62)(42,51)(43,64)(46,60)(47,49)(50,57)(54,55)(56,58)(61,63)

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72341754, #72341886
maximal normal subgroup of	#72341754, #72341886

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407076);