#8197309

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	#47845	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4492 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)(4,6)(5,8)(25,26,30,31)(27,29,33,38)(28,35,37,32)(34,40,39,36);(1,3)(2,5)(4,8)(6,7)(9,10,14,13)(11,17,19,15)(12,16,21,20)(18,23,24,22)(25,26,30,31)(27,29,33,38)(28,32,37,35)(34,36,39,40);(1,2,4,6)(3,5,8,7)(10,13)(15,17)(16,20)(22,23)(26,31)(29,38)(32,35)(36,40);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,18)(13,17)(14,19)(16,22)(20,23)(21,24)(25,27,30,33)(26,29,31,38)(28,34,37,39)(32,36,35,40);(1,3)(2,5)(4,8)(6,7)(9,12,14,21)(10,16,13,20)(11,18,19,24)(15,22,17,23)(25,28)(26,32)(27,34)(29,36)(30,37)(31,35)(33,39)(38,40);(1,2)(3,5)(4,6)(7,8)(9,13,14,10)(11,17,19,15)(12,20,21,16)(18,23,24,22)(25,29)(26,33)(27,31)(28,36)(30,38)(32,39)(34,35)(37,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=g,d^2=gh,e^2=ghi,f^2=ghi,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, 8104505);