#494664

order	512
nilpotency class	4
isoclinism	isoclinic class of #13074
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1893
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#1893
prime radical	#60450	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1285 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)(4,21,24,10)(5,11)(6,27)(7,30)(8,13)(9,18)(12,15)(16,38,46,59)(17,39)(19,41)(20,51)(22,53,25,34)(23,54,28,57)(26,37)(29,56)(31,45,52,49)(32,48)(33,50)(35,55,36,43)(40,42)(44,60,47,64)(58,62,61,63);(1,3,15,42,7,20,29,50,8,19,26,48,5,17,6,18)(2,9,27,39,11,32,37,41,13,33,56,51,30,40,12,14)(4,16,43,62,25,47,23,45,24,46,55,63,22,44,28,49)(10,31,54,60,34,58,35,59,21,52,57,64,53,61,36,38);(1,4)(2,10)(3,16)(5,22)(6,28)(7,25)(8,24)(9,31)(11,34)(12,36)(13,21)(14,38)(15,43)(17,44)(18,49)(19,46)(20,47)(23,29)(26,55)(27,54)(30,53)(32,58)(33,52)(35,37)(39,60)(40,61)(41,59)(42,62)(45,50)(48,63)(51,64)(56,57);(1,5,8,7)(2,11,13,30)(3,17,19,20)(4,22,24,25)(6,26,29,15)(9,32,33,40)(10,34,21,53)(12,27,37,56)(14,39,41,51)(16,44,46,47)(18,48,50,42)(23,43,28,55)(31,58,52,61)(35,57,36,54)(38,60,59,64)(45,62,49,63)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72249778, #72249781, #72249830, …
maximal normal subgroup of	#72249778, #72249781, #72249830, …
maximal quotient of	#72257021, C₂×#494664, #72322923

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401860);