#8934810

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	#47809	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4336 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)(3,5)(4,8)(6,7)(9,10,14,15)(11,16,17,12)(13,21,22,18)(19,20,24,23)(25,26)(27,33)(28,35)(29,32)(30,34)(31,38)(36,40)(37,39);(1,3)(2,5)(4,6)(7,8)(10,15)(11,17)(18,21)(19,24)(25,27,32,31)(26,33,29,38)(28,37,30,36)(34,40,35,39);(1,2)(3,8)(4,5)(6,7)(10,15)(11,17)(13,22)(20,23)(25,28,32,30)(26,34,29,35)(27,36,31,37)(33,39,38,40);(2,7)(3,4)(9,11)(10,16)(12,15)(13,19)(14,17)(18,23)(20,21)(22,24)(25,29,32,26)(27,33,31,38)(28,34,30,35)(36,40,37,39);(1,4,6,3)(2,5,7,8)(9,12,14,16)(10,17,15,11)(13,20,22,23)(18,24,21,19)(25,30,32,28)(26,35,29,34)(27,36,31,37)(33,39,38,40);(1,5)(2,3)(4,7)(6,8)(9,13,14,22)(10,18,15,21)(11,19,17,24)(12,20,16,23)(25,31,32,27)(26,33,29,38)(28,36,30,37)(34,40,35,39)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=gh,d^2=gh,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi

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, 8842006);