#498968

order	512
nilpotency class	4
isoclinism	isoclinic class of #498968
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#486
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#2023
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

921 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)(3,8)(4,18,20,52)(5,23)(6,11)(7,13)(9,21,36,56)(10,26)(12,29)(14,43,44,64)(15,31)(16,47)(17,33)(19,54,50,51)(22,40,28,53)(24,59)(25,37,58,38)(27,41)(30,45,62,61)(32,49)(34,46)(35,55,63,48)(39,60)(42,57);(1,3)(2,8)(4,19,20,50)(5,24)(6,17)(7,15)(9,35,36,63)(10,39)(11,33)(12,34)(13,31)(14,22,44,28)(16,27)(18,51,52,54)(21,48,56,55)(23,59)(25,61,58,45)(26,60)(29,46)(30,37,62,38)(32,42)(40,43,53,64)(41,47)(49,57);(1,4)(2,9)(3,14)(5,25)(6,28)(7,20)(8,30)(10,40)(11,38)(12,37)(13,36)(15,44)(16,48)(17,50)(18,41)(19,46)(21,57)(22,29)(23,52)(24,55)(26,56)(27,58)(31,62)(32,54)(33,63)(34,35)(39,51)(42,53)(43,59)(45,60)(47,64)(49,61);(3,15)(8,31)(14,44)(16,24)(17,46)(19,50)(30,62)(32,39)(33,34)(35,63)(43,64)(45,61)(47,59)(48,55)(49,60)(51,54)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72249757, #72249773, #72249835, …
maximal normal subgroup of	#72249757, #72249773, #72249835, …
maximal quotient of	#72249960, #72323100

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406164);