#8075447

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	#2535	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4204 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)(10,15)(12,22)(16,24)(18,19)(25,26,31,29)(27,34,28,32)(30,33,38,37)(35,40,36,39);(1,2,5,6)(3,7,4,8)(9,10,14,15)(11,16,21,24)(12,17,22,20)(13,18,23,19);(1,3,5,4)(2,7,6,8)(10,15)(12,22)(16,24)(18,19)(25,27,31,28)(26,32,29,34)(30,36,38,35)(33,40,37,39);(1,4,5,3)(2,8,6,7)(9,11,14,21)(10,16,15,24)(12,18,22,19)(13,17,23,20)(25,28,31,27)(26,32,29,34)(30,36,38,35)(33,39,37,40);(9,12)(10,17)(11,19)(13,16)(14,22)(15,20)(18,21)(23,24)(25,29,31,26)(27,34,28,32)(30,37,38,33)(35,40,36,39);(9,13)(10,18)(11,20)(12,16)(14,23)(15,19)(17,21)(22,24)(25,30)(26,33)(27,35)(28,36)(29,37)(31,38)(32,39)(34,40)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=ghi,d^2=gh,e^2=h,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, 7982643);