#148926

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	#1923
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

458 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)(5,6,10,16,11,17,32,46,12,18,33,47,9,15,30,44)(7,19,22,49,21,37,52,45,23,43,31,59,29,13,34,36)(8,14,27,41,28,42,58,66,26,40,57,65,25,39,56,64)(20,48,51,67,50,38,62,63,24,53,55,68,54,35,60,61);(1,2,4,3)(5,7)(6,13)(8,24)(9,29)(10,31)(11,21)(12,23)(14,38)(15,43)(16,45)(17,19)(18,37)(20,26)(22,33)(25,50)(27,51)(28,54)(30,52)(32,34)(35,40)(36,47)(39,48)(41,61)(42,53)(44,49)(46,59)(55,57)(56,60)(58,62)(63,65)(64,68)(66,67);(1,3,4,2)(5,8,9,25,12,26,11,28)(6,14,15,39,18,40,17,42)(7,20,21,50,23,24,29,54)(10,27,30,56,33,57,32,58)(13,35,19,48,37,38,43,53)(16,41,44,64,47,65,46,66)(22,51,52,62,31,55,34,60)(36,61,49,67,45,63,59,68)

Polycyclic presentation:

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

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, 56122);