#9945531

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	#37827	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4297 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=ghi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9852727);