#8422153

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

4562 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8329349);