#8914168

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

4531 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8821364);