#147082

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	#1846
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

528 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)(7,20)(8,26,29,42)(9,17)(10,15)(11,34)(12,19)(13,22)(14,28,41,60)(16,32)(18,31)(21,48,51,35)(23,46)(24,36)(25,47)(27,57,59,43)(30,61,58,40)(33,44)(37,53)(38,55)(39,54)(45,50,65,68)(49,56,67,62)(52,63,66,64);(2,4)(5,7,10,23,11,24,32,54,12,25,33,55,31,53,9,22)(6,13,17,37,18,38,44,47,19,39,16,36,34,46,15,20)(8,21,28,50,30,52,59,67,29,51,60,68,58,66,27,49)(14,35,42,62,43,63,61,65,41,48,26,56,57,64,40,45);(5,8,10,28,11,30,32,59,12,29,33,60,31,58,9,27)(6,14,16,26,18,43,15,40,19,41,17,42,34,57,44,61)(7,21,23,50,24,52,54,67,25,51,55,68,53,66,22,49)(13,35,36,56,38,63,20,45,39,48,37,62,46,64,47,65)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=e,d^2=hi,e^2=h,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,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, 54278);