#8807300

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

4347 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=g,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,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, 8714496);