#7713884

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	#2847	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4292 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=i,e^2=g,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^b=di,d^c=dh,e^c=ei

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, 7621080);