#8431820

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

4869 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8339016);