#8325518

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	#2428	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4273 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=h,d^2=h,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8232714);