#7977459

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	#40520	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4386 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7884655);