#8106053

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	#44022	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4543 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8013249);