#8087941

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×#127	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4098 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)(11,14)(12,16)(13,17)(15,18)(19,21)(20,23)(22,24)(25,26,32,36)(27,33,31,35)(28,30,29,34)(37,40,38,39);(1,2)(3,7)(4,8)(5,6)(25,27,32,31)(26,33,36,35)(28,38,29,37)(30,39,34,40);(1,3,5,4)(2,7,6,8)(9,11,15,12)(10,16,18,14)(13,19,22,20)(17,23,24,21)(25,28,32,29)(26,34,36,30)(27,37,31,38)(33,39,35,40);(9,12,15,11)(10,16,18,14)(13,19,22,20)(17,21,24,23)(25,29,32,28)(26,34,36,30)(27,38,31,37)(33,39,35,40);(9,13,15,22)(10,17,18,24)(11,19,12,20)(14,21,16,23)(25,30,32,34)(26,28,36,29)(27,39,31,40)(33,38,35,37);(1,4,5,3)(2,8,6,7)(9,14,15,16)(10,11,18,12)(13,21,22,23)(17,19,24,20)(25,31,32,27)(26,35,36,33)(28,37,29,38)(30,40,34,39)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gh,d^2=h,e^2=h,f^2=gh,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^b=di,e^c=ei,f^c=fh,e^d=ehi,f^d=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7995137);