#9815264

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	#50641	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4251 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=h,d^2=ghi,e^2=g,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, 9722460);