#7883687

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	#2416	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4362 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)(9,10,15,12)(11,18,16,13)(14,17,23,20)(19,22,24,21)(25,26)(27,28)(29,32)(30,33)(31,34)(35,36)(37,39)(38,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,18)(14,19)(15,16)(17,21)(20,22)(23,24)(25,27,31,29)(26,32,34,28)(30,35,38,37)(33,39,40,36);(10,12)(11,16)(17,20)(19,24)(25,28,31,32)(26,27,34,29)(30,36,38,39)(33,35,40,37);(9,12,15,10)(11,18,16,13)(14,20,23,17)(19,22,24,21);(1,3,4,7)(2,6,5,8)(9,13,15,18)(10,16,12,11)(14,22,23,21)(17,19,20,24)(25,29,31,27)(26,32,34,28)(30,35,38,37)(33,36,40,39);(9,14,15,23)(10,17,12,20)(11,19,16,24)(13,21,18,22)(25,30,31,38)(26,33,34,40)(27,35,29,37)(28,36,32,39)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=hi,d^2=h,e^2=g,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7790883);