#8092016

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	#2541	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4313 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)(11,16)(12,17)(13,18)(14,15)(19,24)(20,21)(22,23)(25,26)(27,33)(28,31)(29,34)(30,32)(35,40)(36,38)(37,39);(1,2,4,5)(3,6,7,8)(25,27,30,28)(26,31,32,33)(29,35,37,36)(34,38,39,40);(1,3,4,7)(2,6,5,8)(10,15)(13,16)(17,23)(21,24)(26,32)(29,37)(31,33)(35,36);(9,11,14,18)(10,16,15,13)(12,20,22,19)(17,21,23,24)(25,28,30,27)(26,33,32,31)(29,35,37,36)(34,38,39,40);(9,12,14,22)(10,17,15,23)(11,19,18,20)(13,21,16,24)(25,26)(27,31)(28,33)(29,34)(30,32)(35,38)(36,40)(37,39);(9,13,14,16)(10,18,15,11)(12,21,22,24)(17,20,23,19)(25,29,30,37)(26,34,32,39)(27,35,28,36)(31,38,33,40)

Polycyclic presentation:

2^9|b^2=g,c^2=gi,d^2=h,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=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, 7999212);