#430462

order	512
nilpotency class	3
isoclinism	isoclinic class of #418484
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
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

1174 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)(5,6,10,18,12,16,11,17)(7,13,23,36,21,38,22,37)(8,24,29,40,27,41,28,14)(9,30,33,15,31,43,32,44)(19,45,47,34,25,54,48,63)(20,49,50,42,52,60,51,35)(26,56,57,61,58,39,59,53)(46,64,67,65,55,62,66,68);(5,7,11,23,12,21,10,22)(6,13,18,37,16,38,17,36)(8,25,29,47,27,19,28,48)(9,20,33,51,31,52,32,50)(14,34,41,54,40,63,24,45)(15,42,30,60,44,35,43,49)(26,46,59,67,58,55,57,66)(39,62,53,65,56,64,61,68);(5,8)(6,14)(7,19)(9,26)(10,29)(11,28)(12,27)(13,34)(15,39)(16,40)(17,41)(18,24)(20,46)(21,25)(22,48)(23,47)(30,61)(31,58)(32,59)(33,57)(35,62)(36,54)(37,45)(38,63)(42,64)(43,53)(44,56)(49,68)(50,67)(51,66)(52,55)(60,65);(2,4)(5,9,10,33,12,31,11,32)(6,15,18,43,16,44,17,30)(7,20,23,50,21,52,22,51)(8,26,29,57,27,58,28,59)(13,35,36,49,38,42,37,60)(14,39,24,53,40,56,41,61)(19,46,47,67,25,55,48,66)(34,62,54,68,63,64,45,65)

Polycyclic presentation:

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

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, 337658);