#9718188

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	#51936	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4164 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=h,d^2=h,e^2=i,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, 9625384);