#8934508

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	#35356	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4511 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=h,e^2=gh,f^2=g,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, 8841704);