#7981477

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	#2416	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4314 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=h,d^2=gi,e^2=gh,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, 7888673);