#109067

order	512
nilpotency class	3
isoclinism	isoclinic class of #8794
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#382
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#124
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

602 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 44 points:

(1,2)(3,4)(6,10)(7,8)(11,12)(13,14)(15,24,28,38)(16,29)(17,22)(18,33)(19,23)(20,26,35,27)(21,31)(25,39,42,34)(30,36)(32,43)(37,40,44,41);(2,4)(5,6,8,12,9,11,7,10)(13,15)(14,20,23,35)(16,25)(17,26)(18,27)(19,28)(21,34,36,39)(22,24,33,38)(29,37,43,44)(30,40)(31,41)(32,42);(6,11)(10,12)(13,16,18,31,19,32,17,30)(14,21,22,29,23,36,33,43)(15,25,27,41,28,42,26,40)(20,34,24,37,35,39,38,44)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 16263);