#10040578

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	#47458	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4166 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)(9,10,15,17)(11,16,21,24)(12,22,23,18)(13,19,20,14)(25,26)(27,34)(28,32)(29,33)(30,35)(31,37)(36,40)(38,39);(2,3)(6,7)(9,11)(10,16)(12,14)(13,22)(15,21)(17,24)(18,20)(19,23)(25,27)(26,31)(28,36)(29,38)(30,37)(32,39)(33,40)(34,35);(1,3,5,2)(4,7,8,6)(10,17)(11,21)(12,23)(13,20)(25,28,30,29)(26,32,35,33)(27,36,37,38)(31,39,34,40);(1,4)(2,6)(3,7)(5,8)(9,12)(10,18)(11,19)(13,16)(14,21)(15,23)(17,22)(20,24)(25,29,30,28)(26,33,35,32)(27,36,37,38)(31,39,34,40);(9,13)(10,19)(11,18)(12,16)(14,17)(15,20)(21,22)(23,24)(25,27)(26,34)(28,38)(29,36)(30,37)(31,35)(32,39)(33,40);(9,14,15,19)(10,20,17,13)(11,12,21,23)(16,18,24,22)(25,26)(27,31)(28,32)(29,33)(30,35)(34,37)(36,39)(38,40)

Polycyclic presentation:

2^9|a^2=i,c^2=g,d^2=ghi,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, 9947774);