#8464002

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	#42838	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4381 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=gi,e^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

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, 8371198);