#496249

order	512
nilpotency class	4
isoclinism	isoclinic class of #496249
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#21750
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×#41	C₂⁴
prime radical	#60819	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #3500 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

854 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)(3,14,18,40)(4,19,23,51)(5,12,26,43)(6,11)(7,13)(8,17,33,27)(9,21)(10,25,39,60)(15,46)(16,45)(20,52)(22,35,28,61)(24,31)(29,38)(30,54,63,53)(32,58)(34,47,44,57)(36,55)(37,56)(41,59)(42,49)(48,62)(50,64);(1,3)(2,8)(4,20)(5,24)(6,27)(7,18)(9,34)(10,31)(11,40)(12,42)(13,33)(14,38)(15,22)(16,26)(17,29)(19,46)(21,54)(23,50)(25,49)(28,48)(30,36)(32,43)(35,64)(37,44)(39,45)(41,63)(47,59)(51,62)(52,61)(53,56)(55,57)(58,60);(1,4)(2,9)(3,15)(5,21)(6,28)(7,23)(8,30)(10,35)(11,41)(12,19)(13,37)(14,44)(16,47)(17,50)(18,48)(20,27)(22,29)(24,57)(25,59)(26,56)(31,62)(32,52)(33,63)(34,40)(36,38)(39,61)(42,64)(43,51)(45,46)(49,54)(53,58)(55,60);(1,5,7,26)(2,10)(3,16,18,24)(4,21,23,56)(6,25,29,60)(8,31)(9,35)(11,12)(13,39)(14,32)(15,47,48,57)(17,49,27,58)(19,41)(20,53,50,54)(22,55,28,59)(30,62)(33,45)(34,64)(36,51)(37,61)(38,43)(40,42)(44,52)(46,63)

Polycyclic presentation:

2^9|a^2=g,d^2=h,e^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=ce,d^b=dh,e^b=ei,f^b=fi,g^b=gi,e^c=ei,g^c=gh,f^d=fi,g^d=gi,g^f=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403445);