#10480834

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

4579 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10388030);