#147002

order	512
nilpotency class	4
isoclinism	isoclinic class of #12026
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×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	#1835
prime radical	C₂⁴	#135
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

512 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)(8,25,28,41)(9,30)(10,16)(11,34)(12,18)(13,21)(14,27,40,57)(15,32)(17,31)(20,47,50,66)(22,44)(23,38)(24,46)(26,39,29,60)(33,42)(35,62,63,68)(36,53)(37,54)(43,49,65,67)(45,52)(48,55,51,64)(56,59,61,58);(2,4)(5,7,11,23,31,52,32,53,12,24,9,21,10,22,33,54)(6,13,15,19,16,36,34,44,18,38,42,46,17,37,30,45)(8,20,29,51,57,67,58,62,28,50,26,48,27,49,59,68)(14,35,39,43,25,55,56,47,40,63,60,65,41,64,61,66);(5,8,12,28)(6,14)(7,20,24,50)(9,26,11,29)(10,27,31,57)(13,35)(15,39)(16,25)(17,41)(18,40)(19,43)(21,48,23,51)(22,49,52,67)(30,61)(32,58,33,59)(34,56)(36,55)(37,64)(38,63)(42,60)(44,47)(45,66)(46,65)(53,62,54,68)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=eh,d^2=hi,e^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,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, 54198);