#10103214

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

4434 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=hi,d^2=gh,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, 10010410);