#8358742

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	#59043	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4329 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=gh,d^2=i,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8265938);