#10482436

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

4605 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=gh,d^2=i,e^2=g,f^2=gh,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, 10389632);