#148323

order	512
nilpotency class	4
isoclinism	isoclinic class of #13313
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#466
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1808
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

658 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 68 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=dh,d^2=hi,e^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,h^a=hi,c^b=cf,e^c=ei,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55519);