#7979793

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×#547	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4372 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)(9,10,15,18)(11,16,12,19)(13,14,17,24)(20,23,22,21)(25,26,31,34)(27,28,29,32)(30,33,39,40)(35,36,37,38);(1,2)(3,6)(4,5)(7,8)(9,10)(11,19)(12,16)(13,14)(15,18)(17,24)(20,21)(22,23)(25,27,31,29)(26,32,34,28)(30,37,39,35)(33,36,40,38);(1,3,4,7)(2,6,5,8)(9,11,15,12)(10,16,18,19)(13,22,17,20)(14,21,24,23)(25,28,31,32)(26,27,34,29)(30,36,39,38)(33,35,40,37);(9,12,15,11)(10,16,18,19)(13,22,17,20)(14,23,24,21)(25,29,31,27)(26,28,34,32)(30,37,39,35)(33,36,40,38);(9,13)(10,14)(11,20)(12,22)(15,17)(16,23)(18,24)(19,21)(25,28)(26,27)(29,34)(30,38)(31,32)(33,37)(35,40)(36,39);(9,14)(10,17)(11,21)(12,23)(13,18)(15,24)(16,20)(19,22)(25,30)(26,33)(27,35)(28,36)(29,37)(31,39)(32,38)(34,40)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=g,d^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, 7886989);