#8810185

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	#2673	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4239 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=gh,d^2=gh,e^2=gh,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8717381);