#8087821

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	#47207	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4262 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=h,e^2=gh,f^2=gh,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^b=di,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, 7995017);