#8417259

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	#57288	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4199 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=g,d^2=gi,e^2=gi,f^2=h,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, 8324455);