#9848998

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	#52383	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4157 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=ghi,d^2=gh,e^2=ghi,f^2=g,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, 9756194);