#10040497

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	#45467	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4217 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=h,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, 9947693);