#9446285

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

4748 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=h,d^2=gh,e^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,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, 9353481);