#465837

order	512
nilpotency class	3
isoclinism	isoclinic class of #455572
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
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

1003 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:

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

Polycyclic presentation:

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

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, 373033);