#8324804

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	#52249	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4230 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=h,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8232000);