#8930681

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

4491 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)(9,10,14,12)(11,15,17,19)(13,16,21,20)(18,22,23,24)(25,26,30,31)(27,28,33,32)(29,37,38,34)(35,39,40,36);(3,7)(5,8)(10,12)(15,19)(16,20)(22,24)(25,27)(26,28)(29,35)(30,33)(31,32)(34,36)(37,39)(38,40);(1,2)(3,5)(4,6)(7,8)(11,17)(13,21)(15,19)(16,20)(26,31)(27,33)(29,38)(36,39);(9,11,14,17)(10,15,12,19)(13,18,21,23)(16,22,20,24)(25,27)(26,32)(28,31)(29,35)(30,33)(34,39)(36,37)(38,40);(1,3)(2,5)(4,7)(6,8)(9,12,14,10)(11,15,17,19)(13,20,21,16)(18,22,23,24)(25,28,30,32)(26,33,31,27)(29,36,38,39)(34,40,37,35);(1,2)(3,5)(4,6)(7,8)(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,29)(26,34)(27,35)(28,36)(30,38)(31,37)(32,39)(33,40)

Polycyclic presentation:

2^9|a^2=gh,d^2=g,e^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8837877);