#10320422

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	#43373	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4195 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)(9,10,15,17)(11,21,22,16)(12,13,19,18)(14,24,23,20)(25,26,31,30)(27,32,36,35)(28,37,29,33)(34,40,38,39);(2,6)(3,7)(9,11)(10,16)(12,23)(13,24)(14,19)(15,22)(17,21)(18,20)(25,27,31,36)(26,32,30,35)(28,38,29,34)(33,40,37,39);(1,3)(2,4)(5,7)(6,8)(10,17)(11,22)(12,19)(20,24)(25,28,31,29)(26,33,30,37)(27,34,36,38)(32,39,35,40);(1,4)(2,3)(5,8)(6,7)(9,12,15,19)(10,18,17,13)(11,23,22,14)(16,20,21,24)(25,27)(26,32)(28,34)(29,38)(30,35)(31,36)(33,39)(37,40);(9,13,15,18)(10,19,17,12)(11,20,22,24)(14,16,23,21)(25,29,31,28)(26,33,30,37)(27,34,36,38)(32,40,35,39);(9,14,15,23)(10,20,17,24)(11,19,22,12)(13,16,18,21)(25,30,31,26)(27,35,36,32)(28,37,29,33)(34,40,38,39)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=gi,d^2=ghi,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, 10227618);