#8928612

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

4834 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)(6,7)(9,10)(11,16)(12,13)(14,22)(15,17)(18,23)(19,21)(20,24)(25,26)(27,30)(28,37)(29,33)(31,34)(32,36)(35,40)(38,39);(1,2,5,4)(3,7,8,6)(9,11,15,12)(10,16,17,13)(14,19,23,20)(18,24,22,21)(25,27)(26,30)(28,29)(31,35)(32,38)(33,37)(34,40)(36,39);(1,3,5,8)(2,6,4,7)(10,17)(11,12)(14,23)(21,24)(27,35)(28,36)(30,40)(32,37);(1,4,5,2)(3,7,8,6)(9,12,15,11)(10,16,17,13)(14,20,23,19)(18,24,22,21)(25,28)(26,32)(27,29)(30,38)(31,36)(33,40)(34,37)(35,39);(9,13,15,16)(10,12,17,11)(14,21,23,24)(18,20,22,19)(25,29)(26,33)(27,36)(28,35)(30,32)(31,39)(34,38)(37,40);(9,14,15,23)(10,18,17,22)(11,19,12,20)(13,21,16,24)(25,30)(26,27)(28,38)(29,32)(31,40)(33,36)(34,35)(37,39)

Polycyclic presentation:

2^9|b^2=gh,c^2=g,d^2=gh,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi

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, 8835808);