#8756280

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

4595 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)(3,7)(4,8)(5,6)(9,10)(11,17)(12,20)(13,18)(14,15)(16,22)(19,21)(23,24)(26,29)(28,36)(31,33)(34,40);(2,6)(3,8)(10,15)(13,17)(16,20)(21,23)(26,29)(31,33)(32,37)(38,39);(1,3)(2,4)(5,8)(6,7)(10,15)(11,18)(16,20)(19,24)(25,26,30,29)(27,33,35,31)(28,32,36,37)(34,39,40,38);(1,4)(2,3)(5,7)(6,8)(9,11)(10,13)(12,19)(14,18)(15,17)(16,21)(20,23)(22,24)(25,27,30,35)(26,31,29,33)(28,34,36,40)(32,38,37,39);(1,2)(3,4)(5,6)(7,8)(9,12,14,22)(10,16,15,20)(11,19,18,24)(13,21,17,23)(25,28)(26,32)(27,34)(29,37)(30,36)(31,38)(33,39)(35,40);(1,4)(2,3)(5,7)(6,8)(9,13)(10,11)(12,21)(14,17)(15,18)(16,19)(20,24)(22,23)(25,29,30,26)(27,33,35,31)(28,37,36,32)(34,39,40,38)

Polycyclic presentation:

2^9|c^2=gh,d^2=gh,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8663476);