#449489

order	512
nilpotency class	3
isoclinism	isoclinic class of #417700
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3162 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)(7,18)(8,23,26,40)(9,27)(10,32)(11,16)(12,17)(13,21)(14,25,41,34)(15,31)(19,47,51,35)(20,42)(22,37)(24,54,57,64)(28,36)(29,43)(30,44)(33,38)(39,58,59,56)(45,50,62,49)(46,52)(48,66,68,60)(53,61)(55,63,67,65);(1,2)(3,4)(5,7)(6,13)(8,19)(9,28)(10,33)(11,21)(12,22)(14,35)(15,42)(16,37)(17,38)(18,32)(20,29)(23,45)(24,55)(25,50)(26,51)(27,46)(30,52)(31,53)(34,49)(36,43)(39,63)(40,62)(41,47)(44,61)(48,56)(54,60)(57,67)(58,68)(59,65)(64,66);(2,4)(5,8)(6,14)(7,19)(9,24)(10,25)(11,34)(12,26)(13,35)(15,39)(16,40)(17,41)(18,45)(20,48)(21,49)(22,51)(23,32)(27,59)(28,55)(29,56)(30,58)(31,57)(33,50)(36,60)(37,62)(38,47)(42,63)(43,54)(44,64)(46,65)(52,68)(53,67)(61,66);(5,9,12,31)(6,15,17,27)(7,20,22,52)(8,24,26,57)(10,30,11,29)(13,36,38,61)(14,39,41,59)(16,44,32,43)(18,46,37,42)(19,48,51,68)(21,53,33,28)(23,54,40,64)(25,58,34,56)(35,60,47,66)(45,65,62,63)(49,67,50,55)

Polycyclic presentation:

2^9|a^2=hi,d^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,e^b=ei,f^c=fi,h^c=hi,e^d=ei,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, 356685);