#8193902

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	#50087	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4499 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=i,e^2=i,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8101098);