#7931482

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	#51419	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4420 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7838678);