#8125668

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

4362 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=g,d^2=h,f^2=g,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, 8032864);