#183831

order	512
nilpotency class	2
isoclinism	isoclinic class of #2668
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

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

663 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,d^2=e,f^2=h,g^2=i,b^a=be,c^a=ci,d^a=di,c^b=ch,d^c=dhi

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 91027);