#8288398

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	#2403	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4475 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8195594);