#8639703

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4160 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)(3,4,7,8)(9,10)(11,18)(12,20)(13,14)(15,16)(17,23)(19,22)(21,24)(25,26,30,29)(27,31,33,35)(28,32,36,37)(34,38,39,40);(1,3)(2,4)(5,7)(6,8)(9,11,15,14)(10,13,16,18)(12,21,23,19)(17,24,20,22)(27,33)(31,35)(34,39)(38,40);(2,6)(3,7)(10,16)(13,18)(17,20)(22,24)(25,27,30,33)(26,31,29,35)(28,34,36,39)(32,38,37,40);(3,7)(4,8)(9,12,15,23)(10,17,16,20)(11,19,14,21)(13,22,18,24)(25,28)(26,32)(27,34)(29,37)(30,36)(31,38)(33,39)(35,40);(1,2)(3,8)(4,7)(5,6)(9,13)(10,11)(12,22)(14,16)(15,18)(17,19)(20,21)(23,24)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,39)(37,40);(1,4,5,8)(2,7,6,3)(9,14,15,11)(10,13,16,18)(12,19,23,21)(17,24,20,22)(25,29,30,26)(27,35,33,31)(28,32,36,37)(34,38,39,40)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=ghi,d^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,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, 8546899);