#464083

order	512
nilpotency class	3
isoclinism	isoclinic class of #455572
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1399 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)(5,6,12,18)(7,19,23,46)(8,14,27,41)(9,28,31,43)(10,17,11,16)(13,34,37,22)(15,30,44,32)(20,47,51,38)(21,36,54,64)(24,56,50,35)(25,42,58,66)(26,45,33,40)(29,39,59,60)(48,55,63,53)(49,61,68,67)(52,65,57,62);(5,7,10,22,12,23,11,34)(6,13,16,19,18,37,17,46)(8,24,26,51,27,50,33,20)(9,21,32,55,31,54,30,53)(14,38,40,56,41,47,45,35)(15,36,28,48,44,64,43,63)(25,57,59,68,58,52,29,49)(39,65,66,67,60,62,42,61);(2,4)(5,8,12,27)(6,14,18,41)(7,20,23,51)(9,29,31,59)(10,33,11,26)(13,35,37,56)(15,42,44,66)(16,45,17,40)(19,47,46,38)(21,52,54,57)(22,50,34,24)(25,32,58,30)(28,60,43,39)(36,62,64,65)(48,67,63,61)(49,55,68,53);(5,9)(6,15)(7,21)(8,25)(10,32)(11,30)(12,31)(13,36)(14,39)(16,28)(17,43)(18,44)(19,48)(20,49)(22,55)(23,54)(24,57)(26,59)(27,58)(29,33)(34,53)(35,61)(37,64)(38,65)(40,66)(41,60)(42,45)(46,63)(47,62)(50,52)(51,68)(56,67)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=i,e^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,h^b=hi,d^c=de,e^c=ei,g^c=gi,e^d=ei,f^d=fi,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, 371279);