#8417285

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	#52417	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4240 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=ghi,d^2=ghi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,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, 8324481);