#8127647

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	#59179	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4338 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=i,d^2=gh,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, 8034843);