#8088004

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×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4128 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=h,d^2=h,e^2=gh,f^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, 7995200);