#8993501

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	#49825	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4377 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)(3,6)(4,5)(7,8)(9,10,14,18)(11,13,17,15)(12,16,20,24)(19,22,21,23)(25,26,30,29)(27,31,36,34)(28,32,37,38)(33,39,35,40);(1,3,5,7)(2,6,4,8)(9,10)(11,15)(12,16)(13,17)(14,18)(19,23)(20,24)(21,22)(25,27)(26,31)(28,35)(29,34)(30,36)(32,40)(33,37)(38,39);(2,4)(3,7)(11,17)(12,20)(13,15)(16,24)(25,28,30,37)(26,32,29,38)(27,33,36,35)(31,39,34,40);(1,3,5,7)(2,6,4,8)(9,11,14,17)(10,15,18,13)(12,21,20,19)(16,22,24,23)(25,29,30,26)(27,34,36,31)(28,32,37,38)(33,39,35,40);(1,3)(2,6)(4,8)(5,7)(9,12,14,20)(10,16,18,24)(11,19,17,21)(13,22,15,23)(25,28)(26,32)(27,35)(29,38)(30,37)(31,40)(33,36)(34,39);(1,4,5,2)(3,8,7,6)(9,13,14,15)(10,17,18,11)(12,22,20,23)(16,21,24,19)(25,27)(26,31)(28,35)(29,34)(30,36)(32,40)(33,37)(38,39)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=hi,d^2=gi,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,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, 8900697);