#9844025

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	#46036	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4215 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)(11,12)(13,16)(14,22)(15,18)(17,23)(19,24)(20,21)(25,26,31,34)(27,35,37,30)(28,32,29,33)(36,40,39,38);(1,2,5,7)(3,4,8,6)(9,10)(11,12)(13,16)(14,17)(15,18)(19,20)(21,24)(22,23)(25,27)(26,30)(28,36)(29,39)(31,37)(32,38)(33,40)(34,35);(1,3)(2,6)(4,7)(5,8)(9,11,15,13)(10,16,18,12)(14,19,23,21)(17,24,22,20)(25,28,31,29)(26,32,34,33)(27,36,37,39)(30,38,35,40);(9,12,15,16)(10,11,18,13)(14,20,23,24)(17,19,22,21)(25,26)(27,30)(28,32)(29,33)(31,34)(35,37)(36,38)(39,40);(1,4,5,6)(2,3,7,8)(9,13,15,11)(10,16,18,12)(14,19,23,21)(17,20,22,24)(25,29,31,28)(26,33,34,32)(27,36,37,39)(30,38,35,40);(1,3)(2,4)(5,8)(6,7)(9,14,15,23)(10,17,18,22)(11,19,13,21)(12,20,16,24)(25,30)(26,27)(28,38)(29,40)(31,35)(32,36)(33,39)(34,37)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=gh,d^2=h,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, 9751221);