#191326

order	512
nilpotency class	3
isoclinism	isoclinic class of #1812
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#117
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2283
prime radical	C₂⁴	#117
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

606 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,d^2=ei,e^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,g^b=gh,g^c=gh

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, 98522);