#9947303

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4306 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,d^2=ghi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9854499);