#146429

order	512
nilpotency class	4
isoclinism	isoclinic class of #12026
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#427
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1808
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

880 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)(7,19)(8,25,27,14)(9,17)(10,15)(11,33)(12,16)(13,21)(18,30)(20,46,48,43)(22,44)(23,54)(24,45)(26,57,60,39)(28,56,61,38)(29,40,59,58)(31,42)(32,41)(34,47,55,67)(35,52)(36,51)(37,53)(49,63,68,64)(50,62,66,65);(2,4)(5,7,10,22,11,23,31,52,12,24,32,53,30,51,9,21)(6,13,17,36,18,37,41,45,16,35,42,54,33,44,15,19)(8,20,28,49,29,50,60,67,27,48,61,68,59,66,26,47)(14,34,39,62,40,63,56,46,25,55,57,65,58,64,38,43);(5,8)(6,14)(7,20)(9,26)(10,28)(11,29)(12,27)(13,34)(15,38)(16,25)(17,39)(18,40)(19,43)(21,47)(22,49)(23,50)(24,48)(30,59)(31,60)(32,61)(33,58)(35,55)(36,62)(37,63)(41,56)(42,57)(44,64)(45,46)(51,66)(52,67)(53,68)(54,65)

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=hi,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=ghi,h^a=hi,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, 53625);