#8077403

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×#560	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4430 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=h,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, 7984599);