#8437830

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	#51207	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4328 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)(3,7,8,4)(9,10,16,14)(11,21,23,17)(12,19,13,18)(15,20,24,22)(25,26)(27,37)(28,33)(29,34)(30,35)(31,36)(32,38)(39,40);(1,3)(2,4)(5,8)(6,7)(9,11,16,23)(10,17,14,21)(12,22,13,20)(15,19,24,18)(27,38)(28,39)(32,37)(33,40);(9,12,16,13)(10,18,14,19)(11,22,23,20)(15,21,24,17)(25,27)(26,32)(28,35)(29,39)(30,40)(31,38)(33,34)(36,37);(2,6)(4,7)(9,13,16,12)(10,19,14,18)(11,22,23,20)(15,21,24,17)(25,28,31,39)(26,33,36,40)(27,35,38,29)(30,32,34,37);(1,3)(2,4)(5,8)(6,7)(9,14,16,10)(11,21,23,17)(12,19,13,18)(15,22,24,20)(25,29,31,35)(26,34,36,30)(27,39,38,28)(32,33,37,40);(1,4)(2,3)(5,7)(6,8)(9,15,16,24)(10,20,14,22)(11,19,23,18)(12,21,13,17)(25,30,31,34)(26,35,36,29)(27,40,38,33)(28,37,39,32)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=hi,d^2=h,e^2=h,f^2=h,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, 8345026);