#8341871

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	#39375	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4393 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=g,e^2=hi,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, 8249067);