#7647161

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4151 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:

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

Polycyclic presentation:

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

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, 7554357);