#501651

order	512
nilpotency class	4
isoclinism	isoclinic class of #501052
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#472	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

639 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)(5,6,10,15,29,42,61,56,9,14,28,41,11,16,31,44)(7,17,20,47,21,37,53,67,19,12,33,35,22,36,54,66)(8,23,26,45,32,63,62,39,25,57,60,40,27,43,30,13)(18,48,50,65,24,58,59,34,49,64,68,38,51,55,52,46);(1,2,4,3)(5,7,9,19)(6,12,14,17)(8,24,25,51)(10,20,28,33)(11,22,29,21)(13,38,39,65)(15,35,41,47)(16,37,42,36)(18,32,49,27)(23,55,57,58)(26,59,60,52)(30,50,62,68)(31,54,61,53)(34,45,46,40)(43,64,63,48)(44,67,56,66);(5,8,9,25)(6,13,14,39)(7,18,19,49)(10,30,28,62)(11,32,29,27)(12,34,17,46)(15,43,41,63)(16,45,42,40)(20,52,33,59)(21,51,22,24)(23,56,57,44)(26,61,60,31)(35,58,47,55)(36,65,37,38)(48,67,64,66)(50,53,68,54);(1,3,4,2)(6,14)(12,17)(13,39)(15,41)(16,42)(23,57)(34,46)(35,47)(36,37)(38,65)(40,45)(43,63)(44,56)(48,64)(55,58)(66,67)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408847);