#8649013

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	#39416	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4512 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:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=hi,d^2=hi,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8556209);