#146430

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	#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

864 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)(5,6,10,17,12,16,32,19)(7,20,23,48,25,47,55,38)(8,26,28,45,31,14,30,41)(9,18,11,15,33,44,34,43)(13,24,37,56,36,57,39,22)(21,49,51,66,54,46,53,64)(27,60,61,40,62,42,29,59)(35,52,63,50,58,67,65,68);(2,4)(5,7,11,24,10,23,33,56,12,25,34,57,32,55,9,22)(6,13,18,38,19,39,43,47,16,36,44,48,17,37,15,20)(8,21,29,52,30,53,62,68,31,54,61,67,28,51,27,50)(14,35,42,64,41,63,59,49,26,58,60,66,45,65,40,46);(5,8)(6,14)(7,21)(9,27)(10,30)(11,29)(12,31)(13,35)(15,40)(16,26)(17,45)(18,42)(19,41)(20,46)(22,50)(23,53)(24,52)(25,54)(28,32)(33,62)(34,61)(36,58)(37,65)(38,64)(39,63)(43,59)(44,60)(47,49)(48,66)(51,55)(56,68)(57,67)

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=hi,f^2=i,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, 53626);