#455575

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
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

1108 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,e^2=i,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,h^a=hi,c^b=cg,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,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, 362771);