#8932176

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	#50285	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4214 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)(3,7)(4,5)(6,8)(9,10,15,18)(11,14,13,17)(12,21,23,16)(19,22,20,24)(25,26,31,35)(27,29,33,40)(28,34,30,32)(36,39,37,38);(2,5)(6,7)(9,11,15,13)(10,14,18,17)(12,20,23,19)(16,22,21,24)(25,27)(26,29)(28,36)(30,37)(31,33)(32,38)(34,39)(35,40);(1,2)(3,6)(4,5)(7,8)(9,12,15,23)(10,16,18,21)(11,19,13,20)(14,24,17,22)(25,28,31,30)(26,32,35,34)(27,36,33,37)(29,38,40,39);(1,3,4,8)(2,6,5,7)(9,12)(10,16)(11,20)(13,19)(14,22)(15,23)(17,24)(18,21);(9,13,15,11)(10,17,18,14)(12,20,23,19)(16,22,21,24)(25,29)(26,33)(27,35)(28,38)(30,39)(31,40)(32,37)(34,36);(9,14,15,17)(10,11,18,13)(12,22,23,24)(16,20,21,19)(25,30,31,28)(26,34,35,32)(27,37,33,36)(29,39,40,38)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8839372);