#459441

order	512
nilpotency class	3
isoclinism	isoclinic class of #456501
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
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

1114 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 366637);