#8418297

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	#44957	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4508 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,d^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,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, 8325493);