#147140

order	512
nilpotency class	3
isoclinism	isoclinic class of B(4,𝔽₂)
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#120
commutator	C₂³	C₂×C₄×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1850
prime radical	C₂⁴	C₂²×C₈
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

530 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54336);