#8132458

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	#39735	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4334 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)(9,10,14,16)(11,18,13,15)(12,20,22,17)(19,23,21,24)(25,26)(27,34)(28,32)(29,31)(30,35)(33,38)(36,40)(37,39);(1,2)(3,6)(4,5)(7,8)(9,11,14,13)(10,15,16,18)(12,21,22,19)(17,24,20,23)(25,27,30,29)(26,31,35,34)(28,36,38,37)(32,39,33,40);(2,5)(6,8)(10,16)(12,22)(15,18)(19,21)(25,28,30,38)(26,32,35,33)(27,36,29,37)(31,39,34,40);(1,2)(3,6)(4,5)(7,8)(9,12,14,22)(10,17,16,20)(11,19,13,21)(15,23,18,24)(25,28)(26,33)(27,36)(29,37)(30,38)(31,40)(32,35)(34,39);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,17)(13,18)(14,16)(19,23)(20,22)(21,24)(25,26)(27,31)(28,33)(29,34)(30,35)(32,38)(36,40)(37,39);(1,3,4,7)(2,6,5,8)(9,13,14,11)(10,18,16,15)(12,21,22,19)(17,24,20,23)(25,29,30,27)(26,34,35,31)(28,37,38,36)(32,40,33,39)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gh,d^2=h,f^2=gi,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, 8039654);