#10320335

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	#40801	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4188 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=gi,d^2=ghi,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dgi,e^b=eghi,d^c=dghi,e^d=egi,f^e=fgi

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, 10227531);