#7853408

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	#43912	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4341 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7760604);