#423175

order	512
nilpotency class	3
isoclinism	isoclinic class of #419839
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1279 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:

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

Polycyclic presentation:

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

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, 330371);