#8928625

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

4989 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=hi,e^2=g,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, 8835821);