#183804

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

665 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)(9,10,13,12,14,16,22,21)(11,15,19,20,18,23,24,17)(25,26,29,33,30,34,38,40)(27,31,28,36,35,39,37,32);(2,5)(6,8)(9,11)(10,15)(12,20)(13,19)(14,18)(16,23)(17,21)(22,24)(25,27,29,28,30,35,38,37)(26,31,33,36,34,39,40,32);(9,12,13,16,14,21,22,10)(11,17,19,15,18,20,24,23)(25,28,29,35,30,37,38,27)(26,32,33,31,34,36,40,39);(1,3,4,7)(2,6,5,8)(11,18)(15,23)(17,20)(19,24)(27,35)(28,37)(31,39)(32,36)

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=ch,c^b=ch,d^b=dh,d^c=di

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, 91000);