#8097060

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	#53116	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4490 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:

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

Polycyclic presentation:

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

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, 8004256);