#8321993

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	#42476	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4374 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:

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

Polycyclic presentation:

2^9|b^2=h,d^2=gh,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8229189);