#8096430

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	#2703	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4240 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:

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

Polycyclic presentation:

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

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, 8003626);