#7642643

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

4360 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)(12,20)(15,22)(18,24)(21,23)(25,26,30,34)(27,33,29,31)(28,32,38,36)(35,40,37,39);(1,2)(3,7)(4,8)(5,6)(9,10,13,16)(11,14,19,17)(12,15,20,22)(18,21,24,23);(1,3,5,4)(2,7,6,8)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26)(27,31)(28,36)(29,33)(30,34)(32,38)(35,40)(37,39);(25,27,30,29)(26,31,34,33)(28,35,38,37)(32,39,36,40);(1,4,5,3)(2,8,6,7)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,28)(26,32)(27,35)(29,37)(30,38)(31,39)(33,40)(34,36);(9,12)(10,15)(11,18)(13,20)(14,21)(16,22)(17,23)(19,24)(25,29,30,27)(26,33,34,31)(28,37,38,35)(32,40,36,39)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=g,d^2=h,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^c=ehi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7549839);