#7657254

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	#41994	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4374 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=hi,d^2=g,e^2=gi,f^2=i,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, 7564450);