#7656812

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	#34593	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4604 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)(11,18)(12,17)(13,14)(15,19)(16,22)(20,24)(21,23)(25,26,30,28)(27,33,35,31)(29,32,37,36)(34,39,40,38);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,23)(13,21)(14,20)(15,22)(17,24)(18,19);(9,12,15,14)(10,17,19,13)(11,20,22,23)(16,21,18,24)(25,27,30,35)(26,31,28,33)(29,34,37,40)(32,38,36,39);(9,13)(10,12)(11,21)(14,19)(15,17)(16,23)(18,20)(22,24);(9,11)(10,18)(12,20)(13,21)(14,23)(15,22)(16,19)(17,24)(25,28,30,26)(27,33,35,31)(29,32,37,36)(34,38,40,39);(1,3,4,7)(2,6,5,8)(9,14,15,12)(10,13,19,17)(11,20,22,23)(16,21,18,24)(25,29)(26,32)(27,34)(28,36)(30,37)(31,38)(33,39)(35,40)

Polycyclic presentation:

2^9|a^2=h,c^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, 7564008);