#8912244

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

4668 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)(3,4)(5,6)(7,8)(9,10)(11,17)(12,20)(13,19)(14,15)(16,21)(18,23)(22,24)(25,26,30,33)(27,31,34,28)(29,32,38,40)(35,37,39,36);(2,6)(3,7)(9,11,14,19)(10,13,15,17)(12,18,21,22)(16,23,20,24)(25,27)(26,31)(28,33)(29,35)(30,34)(32,37)(36,40)(38,39);(1,3,5,7)(2,4,6,8)(10,15)(12,21)(13,17)(18,22)(27,34)(28,31)(35,39)(36,37);(1,4)(2,7)(3,6)(5,8)(9,12)(10,16)(11,18)(13,23)(14,21)(15,20)(17,24)(19,22)(25,28,30,31)(26,27,33,34)(29,37,38,36)(32,39,40,35);(9,11)(10,17)(12,22)(13,15)(14,19)(16,23)(18,21)(20,24)(25,29)(26,32)(27,35)(28,36)(30,38)(31,37)(33,40)(34,39);(9,13,14,17)(10,11,15,19)(12,23,21,24)(16,18,20,22)(25,27)(26,31)(28,33)(29,35)(30,34)(32,37)(36,40)(38,39)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=gh,d^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8819440);