#7647130

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

4167 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=ghi,d^2=i,e^2=hi,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, 7554326);