#9717480

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	#47839	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4211 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=g,d^2=g,e^2=ghi,f^2=i,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,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, 9624676);