#7971824

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

4261 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)(9,10,14,16)(11,17,12,15)(13,18,21,23)(19,24,20,22)(26,31)(27,35)(33,36)(34,39);(1,2)(3,7)(4,8)(5,6)(9,11,14,12)(10,15,16,17)(13,20,21,19)(18,24,23,22)(25,26)(27,32)(28,33)(29,39)(30,31)(34,40)(35,38)(36,37);(1,3,5,4)(2,7,6,8)(10,16)(11,12)(18,23)(19,20)(26,31)(28,37)(32,38)(34,39);(9,10)(11,15)(12,17)(13,18)(14,16)(19,22)(20,24)(21,23)(25,27)(26,32)(28,34)(29,36)(30,35)(31,38)(33,40)(37,39);(1,3,5,4)(2,7,6,8)(9,12,14,11)(10,17,16,15)(13,19,21,20)(18,22,23,24)(25,28,30,37)(26,33,31,36)(27,34,35,39)(29,32,40,38);(1,4,5,3)(2,8,6,7)(9,13,14,21)(10,18,16,23)(11,19,12,20)(15,22,17,24)(25,29,30,40)(26,34,31,39)(27,36,35,33)(28,38,37,32)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,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, 7879020);