#7972720

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

4302 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)(9,10,14,15)(11,17,18,13)(12,20,21,16)(19,23,24,22)(25,26)(27,32)(28,34)(29,30)(31,33)(35,38)(36,37)(39,40);(1,2,5,6)(3,7,4,8)(9,11,14,18)(10,13,15,17)(12,19,21,24)(16,23,20,22)(25,26,31,33)(27,34,28,32)(29,30,39,40)(35,36,37,38);(10,15)(11,18)(12,21)(22,23)(25,27,31,28)(26,32,33,34)(29,37,39,35)(30,36,40,38);(1,3,5,4)(2,7,6,8)(9,10)(11,13)(12,16)(14,15)(17,18)(19,23)(20,21)(22,24)(25,28,31,27)(26,32,33,34)(29,37,39,35)(30,38,40,36);(1,4,5,3)(2,8,6,7)(9,12)(10,16)(11,19)(13,23)(14,21)(15,20)(17,22)(18,24)(25,29,31,39)(26,30,33,40)(27,35,28,37)(32,38,34,36);(9,13)(10,11)(12,22)(14,17)(15,18)(16,24)(19,20)(21,23)(25,30)(26,29)(27,36)(28,38)(31,40)(32,37)(33,39)(34,35)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=hi,d^2=gh,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, 7879916);