#8069682

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	#43096	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4526 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)(3,6,4,7)(9,10)(11,14)(12,16)(13,17)(15,18)(19,22)(20,21)(23,24)(25,26,31,30)(27,34,29,32)(28,33,37,38)(35,40,36,39);(1,2)(3,6)(4,7)(5,8)(9,11)(10,14)(12,19)(13,18)(15,17)(16,22)(20,23)(21,24)(25,27,31,29)(26,32,30,34)(28,35,37,36)(33,39,38,40);(11,18)(14,15)(19,24)(22,23)(26,30)(27,29)(33,38)(35,36);(9,10)(11,14)(12,20)(13,17)(15,18)(16,21)(19,23)(22,24)(25,28)(26,33)(27,35)(29,36)(30,38)(31,37)(32,39)(34,40);(1,3,5,4)(2,6,8,7)(9,11)(10,15)(12,19)(13,18)(14,17)(16,23)(20,22)(21,24)(25,29,31,27)(26,34,30,32)(28,36,37,35)(33,40,38,39);(1,4,5,3)(2,7,8,6)(9,12)(10,16)(11,19)(13,21)(14,22)(15,23)(17,20)(18,24)(25,30,31,26)(27,34,29,32)(28,33,37,38)(35,39,36,40)

Polycyclic presentation:

2^9|a^2=g,b^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,e^a=ei,c^b=ci,e^c=ei,f^c=fh,e^d=ehi,f^d=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7976878);