#418538

order	512
nilpotency class	3
isoclinism	isoclinic class of #418535
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1625 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 325734);