#8127053

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

4315 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=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, 8034249);