#7714597

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	#39838	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4220 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=gh,d^2=h,e^2=gi,f^2=g,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^b=di,d^c=dh,e^c=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7621793);