#7720032

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	#2847	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4350 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=g,d^2=g,e^2=h,f^2=i,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=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, 7627228);