#7989646

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×#127	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4319 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)(11,18)(12,13)(14,16)(15,19)(17,22)(20,21)(23,24)(27,33)(31,35)(34,39)(38,40);(1,3,4,8)(2,5,6,7)(9,11)(10,15)(12,20)(13,23)(14,19)(16,18)(17,21)(22,24)(26,29)(31,35)(32,36)(38,40);(10,16)(11,19)(12,22)(21,23)(25,26,30,29)(27,31,33,35)(28,36,37,32)(34,40,39,38);(9,12)(10,17)(11,20)(13,16)(14,22)(15,21)(18,23)(19,24)(25,27,30,33)(26,31,29,35)(28,34,37,39)(32,38,36,40);(9,11)(10,15)(12,20)(13,23)(14,19)(16,18)(17,21)(22,24)(25,28,30,37)(26,32,29,36)(27,34,33,39)(31,38,35,40);(9,13,14,17)(10,12,16,22)(11,21,19,23)(15,24,18,20)(25,29,30,26)(27,35,33,31)(28,32,37,36)(34,38,39,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=hi,d^2=hi,e^2=hi,f^2=i,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, 7896842);