#511760

order	512
nilpotency class	4
isoclinism	isoclinic class of #511755
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#3584	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3095 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 68 points:

(1,2)(3,4)(5,6,12,18)(7,19,23,48)(8,24,28,45)(9,15,11,17)(10,30,33,43)(13,35,39,22)(14,25,41,27)(16,29,44,32)(20,49,53,65)(21,47,34,42)(26,57,60,40)(31,37,61,38)(36,58,64,68)(46,50,67,52)(51,54,62,63)(55,56,66,59);(5,7)(6,13)(8,20)(9,21)(10,31)(11,34)(12,23)(14,36)(15,37)(16,42)(17,38)(18,39)(19,30)(22,32)(24,54)(25,50)(26,58)(27,52)(28,53)(29,35)(33,61)(40,65)(41,64)(43,48)(44,47)(45,63)(46,55)(49,57)(51,59)(56,62)(60,68)(66,67);(5,8)(6,14)(7,20)(9,25)(10,26)(11,27)(12,28)(13,36)(15,24)(16,40)(17,45)(18,41)(19,46)(21,50)(22,51)(23,53)(29,56)(30,55)(31,58)(32,59)(33,60)(34,52)(35,62)(37,54)(38,63)(39,64)(42,65)(43,66)(44,57)(47,49)(48,67)(61,68);(2,4)(5,9,12,11)(6,15,18,17)(7,21,23,34)(8,25,28,27)(10,29,33,32)(13,37,39,38)(14,24,41,45)(16,30,44,43)(19,47,48,42)(20,50,53,52)(22,31,35,61)(26,56,60,59)(36,54,64,63)(40,55,57,66)(46,49,67,65)(51,58,62,68)

Polycyclic presentation:

2^9|a^2=i,d^2=i,e^2=hi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,h^a=hi,e^b=eh,h^b=hi,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 418956);