#8424531

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	#2416	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4292 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8331727);