#8934891

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	#49741	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4356 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)(9,10)(11,15)(12,21)(13,18)(14,19)(16,20)(17,23)(22,24)(25,26,31,32)(27,33,29,28)(30,38,39,34)(35,36,37,40);(2,3)(6,7)(9,11,14,20)(10,15,19,16)(12,22,23,18)(13,21,24,17)(25,27,31,29)(26,28,32,33)(30,35,39,37)(34,36,38,40);(1,3,5,2)(4,7,8,6)(11,20)(13,24)(15,16)(18,22)(26,32)(28,33)(30,39)(35,37);(1,4,5,8)(2,6,3,7)(9,11)(10,16)(12,22)(13,21)(14,20)(15,19)(17,24)(18,23)(25,28)(26,27)(29,32)(30,36)(31,33)(34,35)(37,38)(39,40);(9,12,14,23)(10,17,19,21)(11,18,20,22)(13,15,24,16)(25,29,31,27)(26,33,32,28)(30,37,39,35)(34,40,38,36);(9,13)(10,18)(11,21)(12,15)(14,24)(16,23)(17,20)(19,22)(25,30,31,39)(26,34,32,38)(27,35,29,37)(28,36,33,40)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,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, 8842087);