#8994324

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	#44282	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4350 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=gi,d^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,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, 8901520);