#7877828

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×#553	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4309 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)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26)(27,32)(28,31)(29,33)(30,35)(34,38)(36,39)(37,40);(1,3,4,8)(2,5,6,7)(9,10)(11,14)(12,15)(13,16)(17,19)(18,22)(20,23)(21,24)(25,27)(26,30)(28,34)(29,35)(31,37)(32,33)(36,40)(38,39);(9,10,13,16)(11,14,19,17)(12,15,21,24)(18,22,20,23)(25,26)(27,32)(28,31)(29,33)(30,35)(34,38)(36,39)(37,40);(9,10)(11,14)(12,15)(13,16)(17,19)(18,22)(20,23)(21,24)(25,28,29,36)(26,31,33,39)(27,34,35,40)(30,37,32,38);(9,11)(10,14)(12,20)(13,19)(15,23)(16,17)(18,21)(22,24)(25,26)(27,30)(28,31)(29,33)(32,35)(34,37)(36,39)(38,40);(9,12,13,21)(10,15,16,24)(11,18,19,20)(14,22,17,23)(25,27)(26,32)(28,34)(29,35)(30,33)(31,38)(36,40)(37,39)

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=h,d^2=hi,f^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,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, 7785024);