#147122

order	512
nilpotency class	3
isoclinism	isoclinic class of #471
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	#121
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1869
prime radical	C₂⁴	#131
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

540 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:

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

Polycyclic presentation:

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

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, 54318);