#8193850

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	#55312	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4539 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=h,d^2=hi,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8101046);