#418517

order	512
nilpotency class	3
isoclinism	isoclinic class of #417622
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
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

1636 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)(7,19)(8,25)(9,30)(10,18)(11,16)(12,17)(13,22)(14,28)(15,33)(20,35)(21,36)(23,37)(24,38)(26,55)(27,42)(29,41)(31,61)(32,45)(34,44)(39,50)(40,59)(43,51)(46,49)(47,64)(48,53)(52,60)(54,58)(56,63)(57,65)(62,67)(66,68);(5,7,12,24)(6,13,17,37)(8,20,29,50)(9,31,34,51)(10,23,11,22)(14,39,42,35)(15,43,45,61)(16,38,18,19)(21,33,52,32)(25,53,41,46)(26,56,59,67)(27,49,28,48)(30,60,44,36)(40,64,55,66)(47,58,68,57)(54,62,65,63);(5,8)(6,14)(7,20)(9,26)(10,27)(11,28)(12,29)(13,35)(15,40)(16,41)(17,42)(18,25)(19,46)(21,47)(22,48)(23,49)(24,50)(30,54)(31,56)(32,57)(33,58)(34,59)(36,62)(37,39)(38,53)(43,66)(44,65)(45,55)(51,67)(52,68)(60,63)(61,64);(2,4)(5,9,11,33,12,34,10,32)(6,15,18,30,17,45,16,44)(7,21,23,51,24,52,22,31)(8,26,28,58,29,59,27,57)(13,36,38,61,37,60,19,43)(14,40,25,54,42,55,41,65)(20,47,49,67,50,68,48,56)(35,62,53,64,39,63,46,66)

Polycyclic presentation:

2^9|b^2=i,d^2=ei,e^2=i,f^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,h^b=hi,f^c=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, 325713);