#8424981

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

4401 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:

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

Polycyclic presentation:

2^9|c^2=ghi,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,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, 8332177);