#8388148

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

4542 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8295344);