#110667

order	512
nilpotency class	3
isoclinism	isoclinic class of #9303
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#123
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

740 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,d^2=hi,e^2=h,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gh,d^b=di,e^b=ei,f^b=fi,d^c=di

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 17863);