#10216754

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	#43193	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4315 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=h,d^2=i,e^2=gi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

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, 10123950);