#7641999

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	C₂²×#41	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4338 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)(11,18)(12,15)(13,21)(14,17)(16,22)(19,23)(20,24)(25,26,31,32)(27,34,30,33)(28,37,39,29)(35,36,38,40);(1,2)(3,7)(4,8)(5,6)(9,11,14,12)(10,15,17,18)(13,19,22,20)(16,23,21,24)(26,32)(29,37)(33,34)(36,40);(1,3,5,4)(2,7,6,8)(9,10)(11,18)(12,15)(13,16)(14,17)(19,24)(20,23)(21,22)(25,27,31,30)(26,33,32,34)(28,35,39,38)(29,36,37,40);(1,3,5,4)(2,7,6,8)(9,12,14,11)(10,15,17,18)(13,20,22,19)(16,23,21,24)(25,28)(26,29)(27,35)(30,38)(31,39)(32,37)(33,36)(34,40);(1,3,5,4)(2,7,6,8)(9,10)(11,18)(12,15)(13,16)(14,17)(19,24)(20,23)(21,22)(25,29)(26,28)(27,36)(30,40)(31,37)(32,39)(33,35)(34,38);(1,4,5,3)(2,8,6,7)(9,13,14,22)(10,16,17,21)(11,19,12,20)(15,23,18,24)(25,30,31,27)(26,34,32,33)(28,38,39,35)(29,40,37,36)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=gi,d^2=gh,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^b=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, 7549195);