#7764873

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	#59315	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4248 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)(11,17)(12,15)(13,18)(14,16)(19,24)(20,22)(21,23)(25,26,31,33)(27,32,35,37)(28,34,36,40)(29,38,39,30);(1,2)(3,7)(4,8)(5,6)(9,11,14,12)(10,15,16,17)(13,19,21,20)(18,22,23,24)(25,27,31,35)(26,32,33,37)(28,29,36,39)(30,34,38,40);(10,16)(15,17)(18,23)(22,24)(26,33)(27,35)(28,36)(30,38);(9,10)(11,15)(12,17)(13,18)(14,16)(19,22)(20,24)(21,23)(25,28)(26,34)(27,29)(30,37)(31,36)(32,38)(33,40)(35,39);(1,3,5,4)(2,7,6,8)(9,12,14,11)(10,17,16,15)(13,20,21,19)(18,24,23,22)(25,29)(26,30)(27,36)(28,35)(31,39)(32,34)(33,38)(37,40);(1,4,5,3)(2,8,6,7)(9,13,14,21)(10,18,16,23)(11,19,12,20)(15,22,17,24)(25,30)(26,29)(27,34)(28,37)(31,38)(32,36)(33,39)(35,40)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7672069);