#499125

order	512
nilpotency class	4
isoclinism	isoclinic class of #13436
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2044
commutator, Frattini, cosocle	C₂×#41	C₂⁴
socle	C₂	C₂×#2044
prime radical	#60749	C₂
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

1084 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)(3,15)(4,21)(5,11)(6,28)(7,12)(8,14)(9,18)(10,24)(13,30)(16,44)(17,34)(19,50)(20,40)(22,54)(23,51)(25,52)(26,53)(27,41)(29,43)(31,39)(32,60)(33,48)(35,46)(36,56)(37,58)(38,57)(42,49)(45,61)(47,64)(55,62)(59,63);(1,3)(2,9)(4,22)(5,17)(6,29)(7,18)(8,20)(10,35)(11,15)(12,40)(13,42)(14,34)(16,25)(19,30)(21,44)(23,47)(24,55)(26,45)(27,49)(28,33)(31,48)(32,37)(36,62)(38,63)(39,50)(41,43)(46,58)(51,54)(52,61)(53,64)(56,59)(57,60);(1,4,5,23,8,26,7,25)(2,10,12,37,14,38,11,36)(3,16,18,45,20,47,17,22)(6,24,27,56,30,57,31,58)(9,32,15,35,34,62,40,63)(13,21,41,52,28,53,39,51)(19,46,49,60,29,59,48,55)(33,61,43,44,42,54,50,64);(1,5,8,7)(2,11,14,12)(3,17,20,18)(4,23,26,25)(6,27,30,31)(9,15,34,40)(10,36,38,37)(13,39,28,41)(16,22,47,45)(19,48,29,49)(21,51,53,52)(24,56,57,58)(32,35,62,63)(33,43,42,50)(44,54,64,61)(46,55,59,60)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303731, #72303741, #72303747, …
maximal normal subgroup of	#72303731, #72303741, #72303747, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406321);