#8942453

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

4367 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=gh,d^2=gh,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,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, 8849649);