#10040860

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	#2682	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4153 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=g,d^2=ghi,e^2=h,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, 9948056);