#10040554

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	#2464	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4100 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)(3,4)(5,6)(7,8)(9,10)(11,18)(12,17)(13,15)(14,16)(19,24)(20,22)(21,23)(25,26,30,31)(27,32,37,40)(28,33,38,36)(29,34,39,35);(1,3,6,7)(2,4,5,8)(9,11,14,13)(10,15,16,18)(12,20,21,19)(17,24,23,22)(26,31)(28,38)(32,40)(34,35);(1,4,6,8)(2,7,5,3)(10,16)(12,21)(15,18)(19,20)(25,27,30,37)(26,32,31,40)(28,35,38,34)(29,36,39,33);(1,2,6,5)(3,8,7,4)(9,12)(10,17)(11,19)(13,20)(14,21)(15,22)(16,23)(18,24)(25,28)(26,33)(27,35)(29,32)(30,38)(31,36)(34,37)(39,40);(1,4)(2,3)(5,7)(6,8)(9,10)(11,18)(12,17)(13,15)(14,16)(19,24)(20,22)(21,23)(25,27)(26,32)(28,35)(29,33)(30,37)(31,40)(34,38)(36,39);(1,5,6,2)(3,8,7,4)(9,13,14,11)(10,18,16,15)(12,20,21,19)(17,24,23,22)(25,29)(26,34)(27,36)(28,32)(30,39)(31,35)(33,37)(38,40)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=gh,d^2=i,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9947750);