#7981464

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4187 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=h,d^2=i,e^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, 7888660);