#148674

order	512
nilpotency class	3
isoclinism	isoclinic class of #2018
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#121
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1869
prime radical	C₂⁴	#131
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

504 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55870);