#494881

order	512
nilpotency class	4
isoclinism	isoclinic class of #33657
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#481
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

762 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402077);