#420388

order	512
nilpotency class	3
isoclinism	isoclinic class of #419800
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
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

1289 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:

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

Polycyclic presentation:

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

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, 327584);