#8587747

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	#43893	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4681 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8494943);