#8656224

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4315 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)(3,8)(4,6)(5,7)(9,10,15,12)(11,16,13,17)(14,18,22,20)(19,23,21,24)(25,26,30,29)(27,33,35,31)(28,32,37,36)(34,39,40,38);(2,4)(7,8)(9,11)(10,16)(12,17)(13,15)(14,19)(18,23)(20,24)(21,22)(25,26,30,29)(27,33,35,31)(28,36,37,32)(34,38,40,39);(1,3,6,5)(2,7,4,8)(11,13)(16,17)(19,21)(23,24)(25,27,30,35)(26,31,29,33)(28,34,37,40)(32,38,36,39);(1,2,6,4)(3,7,5,8)(9,12,15,10)(11,17,13,16)(14,18,22,20)(19,23,21,24)(25,28)(26,32)(27,34)(29,36)(30,37)(31,38)(33,39)(35,40);(1,4,6,2)(3,7,5,8)(9,13,15,11)(10,17,12,16)(14,21,22,19)(18,24,20,23)(25,27)(26,31)(28,34)(29,33)(30,35)(32,38)(36,39)(37,40);(1,5,6,3)(2,7,4,8)(9,14,15,22)(10,18,12,20)(11,19,13,21)(16,23,17,24)(25,29,30,26)(27,33,35,31)(28,32,37,36)(34,38,40,39)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8563420);