#9442920

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	#49617	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4178 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gi,d^2=gh,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,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, 9350116);