#146431

order	512
nilpotency class	4
isoclinism	isoclinic class of #12026
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

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	#1807
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

872 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)(7,18)(8,24,26,14)(9,29)(10,32)(11,34)(12,16)(13,20)(15,31)(17,30)(19,45,47,41)(21,36)(22,53)(23,43)(25,57,27,60)(28,39,58,56)(33,40)(35,46,54,48)(37,52)(38,59,55,61)(42,51)(44,50)(49,64,66,65)(62,68,63,67);(2,4)(5,7,10,21,30,50,31,51,12,23,9,20,11,22,33,52)(6,13,15,18,34,42,32,53,16,36,40,43,17,37,29,44)(8,19,27,48,58,66,59,67,26,47,25,46,28,49,61,68)(14,35,38,41,56,63,57,65,24,54,55,45,39,62,60,64);(5,8)(6,14)(7,19)(9,25)(10,27)(11,28)(12,26)(13,35)(15,38)(16,24)(17,39)(18,41)(20,46)(21,48)(22,49)(23,47)(29,60)(30,58)(31,59)(32,57)(33,61)(34,56)(36,54)(37,62)(40,55)(42,63)(43,45)(44,64)(50,66)(51,67)(52,68)(53,65)

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=hi,g^2=hi,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, 53627);