#8901369

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

4521 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gh,d^2=gi,e^2=hi,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, 8808565);