#500673

order	512
nilpotency class	4
isoclinism	isoclinic class of #499908
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#60648	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1138 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 68 points:

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

Polycyclic presentation:

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

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, 407869);