#8914038

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	#2673	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4202 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)(11,18)(12,17)(13,16)(14,15)(19,24)(20,23)(21,22)(25,26,30,33)(27,28,35,31)(29,37,38,32)(34,39,40,36);(1,2)(3,8)(4,6)(5,7)(11,13)(16,18)(19,20)(23,24)(25,27,30,35)(26,31,33,28)(29,34,38,40)(32,39,37,36);(1,3)(2,4)(5,6)(7,8)(9,11,15,13)(10,16,14,18)(12,20,22,19)(17,24,21,23)(25,26)(27,28)(29,32)(30,33)(31,35)(34,36)(37,38)(39,40);(3,6)(4,8)(9,12,15,22)(10,17,14,21)(11,19,13,20)(16,23,18,24)(25,28)(26,27)(29,36)(30,31)(32,34)(33,35)(37,40)(38,39);(1,4,5,8)(2,6,7,3)(9,13,15,11)(10,16,14,18)(12,19,22,20)(17,24,21,23)(25,27)(26,28)(29,34)(30,35)(31,33)(32,36)(37,39)(38,40);(1,2)(3,8)(4,6)(5,7)(9,14,15,10)(11,16,13,18)(12,21,22,17)(19,23,20,24)(25,29,30,38)(26,32,33,37)(27,34,35,40)(28,36,31,39)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gh,d^2=gh,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8821234);