#8891627

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	#47996	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4382 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=g,d^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8798823);