#8425614

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

4332 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=hi,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,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, 8332810);