#8183014

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

4563 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:

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

Polycyclic presentation:

2^9|b^2=hi,c^2=gh,d^2=h,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8090210);