#465853

order	512
nilpotency class	3
isoclinism	isoclinic class of #458129
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2420
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

1000 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 373049);