#423853

order	512
nilpotency class	3
isoclinism	isoclinic class of #418976
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
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

1277 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)(5,6,10,16,12,17,11,34)(7,18,21,37,23,43,22,13)(8,24,27,39,29,53,28,14)(9,15,31,41,33,42,32,61)(19,45,47,51,25,35,48,44)(20,40,30,59,50,65,49,36)(26,52,56,64,58,68,57,38)(46,66,60,67,54,62,55,63);(5,7,12,23)(6,13,17,37)(8,25,29,19)(9,30,33,49)(10,22,11,21)(14,35,39,45)(15,40,42,65)(16,43,34,18)(20,32,50,31)(24,51,53,44)(26,55,58,60)(27,47,28,48)(36,61,59,41)(38,63,64,67)(46,56,54,57)(52,62,68,66);(5,8,11,28,12,29,10,27)(6,14,16,24,17,39,34,53)(7,19,22,48,23,25,21,47)(9,26,32,57,33,58,31,56)(13,35,18,44,37,45,43,51)(15,38,41,52,42,64,61,68)(20,46,49,55,50,54,30,60)(36,62,40,63,59,66,65,67);(2,4)(5,9,11,32,12,33,10,31)(6,15,34,61,17,42,16,41)(7,20,22,49,23,50,21,30)(8,26,28,57,29,58,27,56)(13,36,43,65,37,59,18,40)(14,38,53,68,39,64,24,52)(19,46,48,55,25,54,47,60)(35,62,51,67,45,66,44,63)

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=f,d^2=ei,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=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, 331049);