#8062873

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	#36136	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4223 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gi,d^2=h,e^2=hi,f^2=h,b^a=bg,c^a=ch,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, 7970069);