#8914521

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

4465 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=h,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8821717);