#8892664

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4128 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=h,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8799860);