#506440

order	512
nilpotency class	4
isoclinism	isoclinic class of #500851
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3058
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#28267
prime radical	C₂×#508	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #2960 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

451 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72217287, #72224999, #72225007, …
maximal normal subgroup of	#72217287, #72224999, #72225007, …
maximal quotient of	C₂×#506440

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 413636);