#8464347

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4540 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8371543);