#8011955

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4139 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=h,d^2=h,e^2=hi,f^2=h,b^a=bg,c^a=ch,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^c=fi,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7919151);