#7656406

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	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4114 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=g,d^2=h,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=di,c^b=ci,f^e=fhi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7563602);