#8994011

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

4420 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=ghi,d^2=hi,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,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, 8901207);