#8107563

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	#43097	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4430 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:

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

Polycyclic presentation:

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

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, 8014759);