#7641907

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	#2847	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4438 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=gi,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^b=ehi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7549103);