#8504793

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	#43927	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4166 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)(3,5)(4,7)(6,8)(11,12)(15,16)(19,20)(22,23)(25,26)(27,34)(28,32)(29,38)(30,33)(31,37)(35,36)(39,40);(2,3)(6,7)(9,10,14,18)(11,15,12,16)(13,17,21,24)(19,22,20,23)(25,27,30,37)(26,31,33,34)(28,35,38,40)(29,36,32,39);(1,2,5,3)(4,6,8,7)(9,11,14,12)(10,15,18,16)(13,19,21,20)(17,22,24,23)(25,28,30,38)(26,32,33,29)(27,35,37,40)(31,39,34,36);(1,3,5,2)(4,7,8,6)(9,10)(11,15)(12,16)(13,17)(14,18)(19,22)(20,23)(21,24)(25,29)(26,28)(27,36)(30,32)(31,35)(33,38)(34,40)(37,39);(1,4,5,8)(2,6,3,7)(9,12,14,11)(10,16,18,15)(13,20,21,19)(17,23,24,22)(25,27)(26,31)(28,35)(29,36)(30,37)(32,39)(33,34)(38,40);(9,13,14,21)(10,17,18,24)(11,19,12,20)(15,22,16,23)(25,26)(27,31)(28,32)(29,38)(30,33)(34,37)(35,39)(36,40)

Polycyclic presentation:

2^9|b^2=gh,c^2=ghi,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, 8411989);