#8824722

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

4779 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)(3,6)(4,8)(5,7)(9,10,14,13)(11,17,18,15)(12,20,22,16)(19,23,21,24)(25,26,30,31)(27,29,34,32)(28,36,37,33)(35,40,38,39);(1,2)(3,6)(4,8)(5,7)(10,13)(15,17)(16,20)(23,24)(26,31)(28,37)(29,32)(35,38);(1,3)(2,4)(5,8)(6,7)(11,18)(15,17)(19,21)(23,24)(26,31)(27,34)(33,36)(35,38);(1,3,5,8)(2,6,7,4)(9,11,14,18)(10,15,13,17)(12,21,22,19)(16,24,20,23)(25,27)(26,32)(28,38)(29,31)(30,34)(33,39)(35,37)(36,40);(1,3)(2,4)(5,8)(6,7)(9,12,14,22)(10,16,13,20)(11,19,18,21)(15,23,17,24)(25,28,30,37)(26,33,31,36)(27,35,34,38)(29,39,32,40);(1,4)(2,3)(5,6)(7,8)(9,13,14,10)(11,17,18,15)(12,20,22,16)(19,24,21,23)(25,29,30,32)(26,27,31,34)(28,39,37,40)(33,35,36,38)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8731918);