#10096857

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4130 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

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, 10004053);