#8827825

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4569 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8735021);