#499212

order	512
nilpotency class	4
isoclinism	isoclinic class of #13460
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2058
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#2058
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

769 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)(4,21,26,53)(5,11)(6,13)(7,29)(8,14)(9,18)(10,25,39,57)(12,27)(16,44,47,63)(17,35)(19,33)(20,34)(22,54,45,59)(23,41,24,52)(28,42)(30,37,56,38)(31,40)(32,50,60,55)(36,46,62,64)(43,48)(49,58)(51,61);(1,3,6,18,8,20,5,17)(2,9,11,15,14,35,13,34)(4,22,24,16,26,45,23,47)(7,19,28,49,31,51,27,48)(10,36,37,60,39,62,38,32)(12,33,40,43,42,61,29,58)(21,44,41,54,53,63,52,59)(25,55,30,46,57,50,56,64);(1,4,6,24,8,26,5,23)(2,10,11,37,14,39,13,38)(3,16,18,45,20,47,17,22)(7,30,28,57,31,56,27,25)(9,32,15,36,35,60,34,62)(12,41,40,53,42,52,29,21)(19,50,49,64,51,55,48,46)(33,44,43,54,61,63,58,59);(1,5,8,6)(2,11,14,13)(3,17,20,18)(4,23,26,24)(7,27,31,28)(9,15,35,34)(10,37,39,38)(12,40,42,29)(16,22,47,45)(19,48,51,49)(21,41,53,52)(25,56,57,30)(32,36,60,62)(33,43,61,58)(44,54,63,59)(46,55,64,50)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72249772, #72249773, #72249825, …
maximal normal subgroup of	#72249772, #72249773, #72249825, …
maximal quotient of	#72250189, C₂×#499212, #72322894, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406408);