#495679

order	512
nilpotency class	3
isoclinism	isoclinic class of #453
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₁₆	#120
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₈×#120
prime radical	#3018	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₁₆ ◁ C₂²×C₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

926 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224962, #72225458, #72254591, …
maximal normal subgroup of	#72224962, #72225458, #72320755, …
maximal quotient of	C₁₆×#453

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402875);