#7658214

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	#2404	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4258 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)(6,8)(9,10)(11,16)(12,22)(13,18)(14,17)(15,19)(20,24)(21,23)(25,26,31,35)(27,34,30,32)(28,38,33,29)(36,40,39,37);(1,2)(3,6)(4,5)(7,8)(9,11,15,13)(10,16,19,18)(12,20,14,21)(17,23,22,24)(25,27,31,30)(26,32,35,34)(28,39,33,36)(29,37,38,40);(9,12,15,14)(10,17,19,22)(11,20,13,21)(16,23,18,24)(25,28,31,33)(26,29,35,38)(27,36,30,39)(32,40,34,37);(9,13,15,11)(10,18,19,16)(12,21,14,20)(17,24,22,23)(25,29)(26,33)(27,37)(28,35)(30,40)(31,38)(32,36)(34,39);(9,14,15,12)(10,17,19,22)(11,21,13,20)(16,23,18,24)(25,27,31,30)(26,34,35,32)(28,36,33,39)(29,37,38,40);(1,3,4,7)(2,6,5,8)(9,10)(11,16)(12,17)(13,18)(14,22)(15,19)(20,23)(21,24)(25,30,31,27)(26,32,35,34)(28,39,33,36)(29,40,38,37)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,d^2=hi,e^2=h,f^2=g,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, 7565410);