#8914213

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

4490 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=gh,d^2=gh,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8821409);