#8188116

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
prime radical	#2428	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4360 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,d^2=hi,e^2=g,f^2=g,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, 8095312);