#8509933

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	#45444	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4626 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=gh,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=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, 8417129);