#191243

order	512
nilpotency class	3
isoclinism	isoclinic class of #191243
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#382
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#124
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=e,e^2=i,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=ch,d^b=dhi,g^b=ghi,g^c=ghi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 98439);