#10196152

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

4420 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=hi,e^2=g,f^2=i,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, 10103348);