#8180454

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	#55684	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4444 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)(10,11)(12,19)(16,18)(20,24)(25,26)(27,28)(29,38)(30,33)(31,34)(32,35)(36,39)(37,40);(1,2,5,4)(3,6,8,7)(9,10,14,11)(12,15,19,17)(13,16,21,18)(20,22,24,23)(25,27)(26,29)(28,35)(30,36)(31,37)(32,38)(33,34)(39,40);(2,4)(6,7)(9,10,14,11)(12,15,19,17)(13,18,21,16)(20,23,24,22)(25,28,32,29)(26,27,35,38)(30,39,37,34)(31,33,36,40);(9,11,14,10)(12,17,19,15)(13,18,21,16)(20,23,24,22)(25,29,32,28)(26,27,35,38)(30,34,37,39)(31,33,36,40);(1,3,5,8)(2,6,4,7)(9,12)(10,15)(11,17)(13,20)(14,19)(16,22)(18,23)(21,24)(25,30,32,37)(26,33,35,40)(27,36,38,31)(28,39,29,34);(1,4,5,2)(3,7,8,6)(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,31,32,36)(26,34,35,39)(27,37,38,30)(28,40,29,33)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=g,d^2=g,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8087650);