#498976

order	512
nilpotency class	4
isoclinism	isoclinic class of #13374
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2023
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

923 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)(4,18,21,50)(5,10)(6,26)(7,12)(9,22,35,53)(11,24)(13,25)(14,40,19,51)(15,30)(16,46)(17,33)(20,49,23,37)(27,39,55,34)(28,36)(29,42,58,52)(31,44)(32,45)(38,54)(41,63,43,60)(47,62,56,57)(48,59)(61,64);(1,3)(2,8)(4,19,21,14)(5,15)(6,16)(7,17)(9,29,35,58)(10,30)(11,31)(12,33)(13,32)(18,40,50,51)(20,43,23,41)(22,52,53,42)(24,44)(25,45)(26,46)(27,56,55,47)(28,48)(34,57,39,62)(36,59)(37,60,49,63)(38,61)(54,64);(1,4)(2,9)(3,14)(5,20)(6,27)(7,21)(8,29)(10,34)(11,37)(12,35)(13,39)(15,41)(16,47)(17,19)(18,36)(22,54)(23,25)(24,53)(26,50)(28,55)(30,57)(31,60)(32,62)(33,58)(38,49)(40,59)(42,64)(43,45)(44,52)(46,51)(48,56)(61,63);(1,5,7,25)(2,10,12,13)(3,15,17,45)(4,20,21,23)(6,24,28,54)(8,30,33,32)(9,34,35,39)(11,36,38,26)(14,41,19,43)(16,44,48,64)(18,49,50,37)(22,27,53,55)(29,57,58,62)(31,59,61,46)(40,63,51,60)(42,47,52,56)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72249772, #72249773, #72249827, …
maximal normal subgroup of	#72249772, #72249773, #72249827, …
maximal quotient of	#72250194, C₂×#498976, #72323124

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406172);