#9851250

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	#45693	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4219 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=gi,e^2=ghi,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, 9758446);