#10320419

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

4055 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)(9,10,15,13)(11,19,21,16)(12,22,23,17)(14,24,20,18)(25,26,31,29)(27,32,35,28)(30,37,39,33)(34,36,38,40);(2,6)(3,7)(9,11,15,21)(10,16,13,19)(12,20,23,14)(17,18,22,24)(25,27,31,35)(26,32,29,28)(30,34,39,38)(33,40,37,36);(1,3)(2,4)(5,7)(6,8)(9,12,15,23)(10,17,13,22)(11,14,21,20)(16,24,19,18)(25,28,31,32)(26,27,29,35)(30,36,39,40)(33,34,37,38);(1,4)(2,3)(5,8)(6,7)(9,12)(10,17)(11,20)(13,22)(14,21)(15,23)(16,18)(19,24)(25,29,31,26)(27,32,35,28)(30,37,39,33)(34,40,38,36);(9,13,15,10)(11,16,21,19)(12,22,23,17)(14,24,20,18)(25,27)(26,32)(28,29)(30,38)(31,35)(33,36)(34,39)(37,40);(9,14,15,20)(10,18,13,24)(11,12,21,23)(16,22,19,17)(25,30)(26,33)(27,34)(28,36)(29,37)(31,39)(32,40)(35,38)

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=ghi,d^2=gi,e^2=h,f^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, 10227615);