#146405

order	512
nilpotency class	3
isoclinism	isoclinic class of #1812
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#117
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1752
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×M₃₂ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

884 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)(3,4)(5,6,10,18,12,19,32,47,13,17,11,35,9,16,31,46)(7,20,23,49,24,14,36,39,25,41,33,50,22,38,57,40)(8,26,28,44,29,42,60,64,30,15,34,43,27,45,59,65)(21,51,53,67,55,58,68,63,56,48,54,66,52,37,61,62);(2,4)(5,7,11,23,12,24,31,36,13,25,10,33,9,22,32,57)(6,14,18,40,19,41,47,49,17,38,35,39,16,20,46,50)(8,21,28,54,29,55,60,61,30,56,34,53,27,52,59,68)(15,37,44,62,45,51,64,67,26,58,43,63,42,48,65,66);(5,8)(6,15)(7,21)(9,27)(10,34)(11,28)(12,29)(13,30)(14,37)(16,42)(17,26)(18,44)(19,45)(20,48)(22,52)(23,54)(24,55)(25,56)(31,60)(32,59)(33,53)(35,43)(36,61)(38,58)(39,63)(40,62)(41,51)(46,65)(47,64)(49,67)(50,66)(57,68)

Polycyclic presentation:

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

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, 53601);