#7901381

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	#59110	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4372 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)(9,10,14,16)(11,13,18,15)(12,20,22,17)(19,21,24,23)(25,26)(27,36)(28,34)(29,30)(31,33)(32,35)(37,40)(38,39);(1,3,5,8)(2,6,4,7)(9,11,14,18)(10,15,16,13)(12,19,22,24)(17,21,20,23)(25,27,32,31)(26,33,35,36)(28,39,30,37)(29,38,34,40);(1,2,5,4)(3,6,8,7)(10,16)(11,18)(17,20)(19,24)(25,28,32,30)(26,34,35,29)(27,37,31,39)(33,38,36,40);(1,2,5,4)(3,6,8,7)(9,10)(11,15)(12,17)(13,18)(14,16)(19,21)(20,22)(23,24)(25,29)(26,28)(27,38)(30,35)(31,40)(32,34)(33,39)(36,37);(1,4,5,2)(3,7,8,6)(9,12)(10,17)(11,19)(13,23)(14,22)(15,21)(16,20)(18,24)(25,30,32,28)(26,29,35,34)(27,37,31,39)(33,38,36,40);(9,13,14,15)(10,18,16,11)(12,21,22,23)(17,19,20,24)(25,31,32,27)(26,33,35,36)(28,39,30,37)(29,40,34,38)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=gh,d^2=gi,e^2=gh,f^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=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, 7808577);