#8075341

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	#2688	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4199 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=i,d^2=h,e^2=g,b^a=bg,c^a=ch,e^a=ei,c^b=ci,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, 7982537);