#8424880

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

4412 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8332076);