#420419

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

1636 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=i,c^2=ei,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,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, 327615);