#497451

order	512
nilpotency class	4
isoclinism	isoclinic class of #497451
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂²⋊C₄×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#3415	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1757 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

978 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)(4,9,21,34)(5,23)(6,13)(7,10)(8,17)(11,24)(12,25)(15,40,43,55)(16,39)(18,29)(19,47)(20,48,51,58)(22,32,50,35)(26,36)(27,37)(28,54,56,42)(30,45)(31,46)(33,59,61,49)(38,53)(41,62,63,57)(44,64,52,60);(1,3)(2,8)(4,15)(5,16)(6,18)(7,27)(9,28)(10,30)(11,37)(12,39)(13,31)(14,23)(17,24)(19,25)(20,41)(21,43)(22,52)(26,45)(29,36)(32,57)(33,60)(34,56)(35,62)(38,47)(40,48)(42,49)(44,50)(46,53)(51,63)(54,59)(55,58)(61,64);(1,4)(2,9)(3,15)(5,20)(6,21)(7,22)(8,28)(10,32)(11,33)(12,35)(13,34)(14,40)(16,41)(17,42)(18,43)(19,44)(23,48)(24,49)(25,50)(26,51)(27,52)(29,55)(30,57)(31,56)(36,58)(37,60)(38,61)(39,62)(45,63)(46,54)(47,64)(53,59);(2,10)(3,16)(5,24)(7,25)(8,29)(9,32)(12,13)(14,31)(15,41)(17,19)(18,45)(20,49)(22,50)(23,36)(26,53)(27,46)(28,55)(30,37)(34,35)(39,47)(40,56)(42,44)(43,63)(48,58)(51,59)(52,54)(57,60)(62,64)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319748, #72343953, #72343960
maximal normal subgroup of	#72319748, #72343953, #72343960

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404647);