#7971657

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×#548	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4291 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,d^2=gh,e^2=h,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, 7878853);