#499055

order	512
nilpotency class	4
isoclinism	isoclinic class of #499054
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#481
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1048 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)(7,19,21,40)(8,25,30,60)(9,17,11,15)(10,32,36,46)(13,24,38,22)(14,41,45,28)(16,31,49,35)(20,52,53,66)(23,39,33,47)(26,61,54,65)(27,48,29,59)(34,43,62,44)(37,57,64,63)(42,55,50,68)(51,56,67,58);(1,2)(3,4)(5,7,11,24,12,21,9,22)(6,13,17,40,18,38,15,19)(8,26,27,20,30,54,29,53)(10,33,35,56,36,23,31,58)(14,42,43,37,45,50,44,64)(16,47,32,51,49,39,46,67)(25,52,59,61,60,66,48,65)(28,63,62,68,41,57,34,55);(2,4)(5,8,9,27,12,30,11,29)(6,14,15,43,18,45,17,44)(7,20,22,54,21,53,24,26)(10,34,31,28,36,62,35,41)(13,37,19,50,38,64,40,42)(16,48,46,25,49,59,32,60)(23,57,56,55,33,63,58,68)(39,66,51,65,47,52,67,61);(7,21)(13,38)(19,40)(20,53)(22,24)(23,33)(26,54)(37,64)(39,47)(42,50)(51,67)(52,66)(55,68)(56,58)(57,63)(61,65)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281854, #72281975, #72282164
maximal normal subgroup of	#72281854, #72281975, #72282164

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406251);