#7806381

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	#2428	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4357 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)(9,10,15,19)(11,20,23,16)(12,17,24,14)(13,21,22,18);(1,3,5,8)(2,6,4,7)(9,11)(10,16)(12,21)(13,17)(14,22)(15,23)(18,24)(19,20)(26,31)(27,28)(29,37)(39,40);(9,12,15,24)(10,14,19,17)(11,21,23,18)(13,16,22,20)(25,26)(27,32)(28,33)(29,38)(30,31)(34,37)(35,40)(36,39);(1,2,5,4)(3,6,8,7)(9,12)(10,17)(11,21)(13,16)(14,19)(15,24)(18,23)(20,22)(25,27,30,28)(26,32,31,33)(29,35,37,36)(34,39,38,40);(1,2,5,4)(3,6,8,7)(9,13,15,22)(10,18,19,21)(11,17,23,14)(12,16,24,20)(25,28,30,27)(26,33,31,32)(29,36,37,35)(34,40,38,39);(1,4,5,2)(3,7,8,6)(9,14,15,17)(10,12,19,24)(11,22,23,13)(16,21,20,18)(25,29)(26,34)(27,35)(28,36)(30,37)(31,38)(32,39)(33,40)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,d^2=ghi,e^2=gi,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^b=di,e^b=ei,f^b=fi,d^c=dh,f^c=fi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7713577);