#10107004

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

4401 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)(11,20,14,16)(12,21,22,18)(13,19,24,23)(25,26)(27,35)(28,38)(29,32)(30,34)(31,33)(36,39)(37,40);(1,3,5,7)(2,4,8,6)(9,11,15,14)(10,16,17,20)(12,19,22,23)(13,18,24,21)(25,27,32,31)(26,33,29,35)(28,37,30,36)(34,40,38,39);(1,3)(2,6)(4,8)(5,7)(10,17)(12,22)(16,20)(19,23)(25,28,32,30)(26,34,29,38)(27,36,31,37)(33,39,35,40);(1,3,5,7)(2,4,8,6)(9,12,15,22)(10,18,17,21)(11,19,14,23)(13,16,24,20)(25,29,32,26)(27,35,31,33)(28,34,30,38)(36,39,37,40);(1,3)(2,4)(5,7)(6,8)(9,13,15,24)(10,19,17,23)(11,21,14,18)(12,20,22,16)(25,30,32,28)(26,34,29,38)(27,37,31,36)(33,39,35,40);(1,4)(2,7)(3,8)(5,6)(9,14,15,11)(10,20,17,16)(12,23,22,19)(13,21,24,18)(25,31,32,27)(26,35,29,33)(28,37,30,36)(34,40,38,39)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=ghi,d^2=h,e^2=g,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, 10014200);