#8799147

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	#49376	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4381 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=h,d^2=gi,e^2=i,f^2=h,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, 8706343);