#8339868

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

4318 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8247064);