#9843765

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×#548	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4114 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=g,d^2=h,e^2=i,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9750961);