#8197056

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	#34977	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4485 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)(9,10,16,15)(11,20,14,17)(12,19,13,18)(21,23,22,24)(26,31)(27,28)(34,38)(35,36);(2,3)(6,7)(9,11,16,14)(10,17,15,20)(12,22,13,21)(18,24,19,23)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,40)(37,38);(1,3,5,2)(4,7,8,6)(9,12,16,13)(10,18,15,19)(11,21,14,22)(17,23,20,24)(25,27,30,28)(26,32,31,33)(29,36,37,35)(34,40,38,39);(1,4)(2,6)(3,7)(5,8)(9,13,16,12)(10,19,15,18)(11,22,14,21)(17,24,20,23)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,40)(37,38);(9,14,16,11)(10,20,15,17)(12,22,13,21)(18,24,19,23)(25,28,30,27)(26,33,31,32)(29,36,37,35)(34,40,38,39);(9,15,16,10)(11,20,14,17)(12,19,13,18)(21,24,22,23)(25,29)(26,34)(27,35)(28,36)(30,37)(31,38)(32,39)(33,40)

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=g,d^2=ghi,e^2=h,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, 8104252);