#9848516

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	#43045	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4161 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gi,d^2=g,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, 9755712);