#8504735

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	#2416	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4107 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)(3,7)(4,6)(5,8)(9,10,15,14)(11,16,21,20)(12,22,17,13)(18,24,23,19)(26,28)(31,33)(32,36)(38,39);(1,3,4,8)(2,5,6,7)(9,11,15,21)(10,16,14,20)(12,18,17,23)(13,19,22,24)(25,26,30,28)(27,33,35,31)(29,32,37,36)(34,39,40,38);(9,12)(10,13)(11,18)(14,22)(15,17)(16,19)(20,24)(21,23)(25,27,30,35)(26,31,28,33)(29,34,37,40)(32,38,36,39);(2,6)(5,7)(9,11)(10,16)(12,18)(13,19)(14,20)(15,21)(17,23)(22,24)(25,28,30,26)(27,33,35,31)(29,36,37,32)(34,39,40,38);(1,3)(2,5)(4,8)(6,7)(9,13,15,22)(10,17,14,12)(11,19,21,24)(16,23,20,18);(1,2)(3,5)(4,6)(7,8)(9,14,15,10)(11,20,21,16)(12,22,17,13)(18,24,23,19)(25,29,30,37)(26,32,28,36)(27,34,35,40)(31,38,33,39)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=i,d^2=i,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=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, 8411931);