#8273413

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	#37725	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4367 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:

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

Polycyclic presentation:

2^9|a^2=h,d^2=g,e^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8180609);