#419855

order	512
nilpotency class	3
isoclinism	isoclinic class of #419852
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1540 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:

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

Polycyclic presentation:

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

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, 327051);