#8172616

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
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4220 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

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

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, 8079812);