#8944431

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	#47938	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4399 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=ghi,d^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8851627);