#8467073

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	#45486	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4353 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)(3,4)(5,7)(6,8)(9,10,15,14)(11,16,21,19)(12,17,22,24)(13,23,20,18)(25,26,32,30)(27,36,39,33)(28,40,34,29)(31,35,38,37);(1,3)(2,6)(4,7)(5,8)(9,10,15,14)(11,19,21,16)(12,17,22,24)(13,18,20,23)(25,27)(26,33)(28,37)(29,31)(30,36)(32,39)(34,35)(38,40);(9,11,15,21)(10,16,14,19)(12,20,22,13)(17,18,24,23)(25,28)(26,29)(27,37)(30,40)(31,33)(32,34)(35,39)(36,38);(2,7)(4,6)(9,12)(10,17)(11,20)(13,21)(14,24)(15,22)(16,18)(19,23)(25,29,32,40)(26,34,30,28)(27,38,39,31)(33,37,36,35);(1,3)(2,6)(4,7)(5,8)(9,13,15,20)(10,18,14,23)(11,12,21,22)(16,24,19,17)(25,30,32,26)(27,36,39,33)(28,40,34,29)(31,37,38,35);(1,4,5,6)(2,3,7,8)(9,14,15,10)(11,19,21,16)(12,17,22,24)(13,23,20,18)(25,31,32,38)(26,35,30,37)(27,29,39,40)(28,33,34,36)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=i,d^2=h,e^2=hi,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, 8374269);