#8630857

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	#37824	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4370 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)(3,4)(5,8)(6,7)(9,10,16,21)(11,14,23,17)(12,19,13,18)(15,20,22,24)(26,29)(27,35)(28,38)(34,40);(1,3)(2,4)(5,6)(7,8)(9,11)(10,17)(12,22)(13,15)(14,21)(16,23)(18,24)(19,20)(25,26,31,29)(27,36,35,32)(28,39,38,33)(30,34,37,40);(1,4)(2,5)(3,7)(6,8)(9,12,16,13)(10,18,21,19)(11,22,23,15)(14,20,17,24)(25,27)(26,32)(28,37)(29,36)(30,38)(31,35)(33,40)(34,39);(1,3,6,5)(2,4,7,8)(9,13,16,12)(10,19,21,18)(11,22,23,15)(14,20,17,24)(25,28,31,38)(26,33,29,39)(27,37,35,30)(32,40,36,34);(1,5,6,3)(2,4,7,8)(9,14)(10,11)(12,24)(13,20)(15,18)(16,17)(19,22)(21,23)(25,29,31,26)(27,36,35,32)(28,39,38,33)(30,40,37,34);(1,4)(2,3)(5,7)(6,8)(9,15,16,22)(10,20,21,24)(11,13,23,12)(14,18,17,19)(25,30,31,37)(26,34,29,40)(27,38,35,28)(32,39,36,33)

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=hi,d^2=h,e^2=i,f^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, 8538053);