#10106988

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

4477 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)(9,10,15,17)(11,18,14,16)(12,21,22,13)(19,24,23,20)(25,26)(27,30)(28,33)(29,38)(31,34)(32,37)(35,36)(39,40);(1,3,5,7)(2,4,8,6)(9,11,15,14)(10,16,17,18)(12,19,22,23)(13,24,21,20)(25,27)(26,32)(28,35)(29,36)(30,34)(31,37)(33,40)(38,39);(1,3)(2,6)(4,8)(5,7)(10,17)(12,22)(16,18)(19,23)(25,28,31,38)(26,33,34,29)(27,35,37,39)(30,36,32,40);(1,3,5,7)(2,4,8,6)(9,12,15,22)(10,13,17,21)(11,19,14,23)(16,24,18,20)(25,29)(26,28)(27,36)(30,39)(31,33)(32,35)(34,38)(37,40);(1,3)(2,4)(5,7)(6,8)(9,13)(10,12)(11,20)(14,24)(15,21)(16,23)(17,22)(18,19)(25,26)(27,32)(28,33)(29,38)(30,37)(31,34)(35,40)(36,39);(1,4)(2,7)(3,8)(5,6)(9,14,15,11)(10,18,17,16)(12,23,22,19)(13,20,21,24)(25,30,31,32)(26,27,34,37)(28,36,38,40)(29,39,33,35)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=ghi,d^2=h,f^2=g,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, 10014184);