#8075327

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	#47760	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4409 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gi,d^2=gh,f^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, 7982523);