#9843759

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	#2407	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4244 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,d^2=gh,e^2=gi,f^2=h,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, 9750955);