#8096461

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	#53066	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4349 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8003657);