#7720533

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	C₂²×#41	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4239 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)(9,10,14,11)(12,15,19,17)(13,18,21,16)(20,23,24,22)(25,26,30,28)(27,33,35,31)(29,36,37,32)(34,38,40,39);(1,2)(3,6)(4,5)(7,8)(10,11)(12,19)(16,18)(20,24);(1,3,4,7)(2,6,5,8)(9,11,14,10)(12,15,19,17)(13,18,21,16)(20,22,24,23)(25,27,30,35)(26,31,28,33)(29,34,37,40)(32,38,36,39);(25,28,30,26)(27,31,35,33)(29,36,37,32)(34,38,40,39);(9,12)(10,15)(11,17)(13,20)(14,19)(16,22)(18,23)(21,24)(25,29,30,37)(26,32,28,36)(27,34,35,40)(31,38,33,39);(9,13,14,21)(10,16,11,18)(12,20,19,24)(15,22,17,23)

Polycyclic presentation:

2^9|a^2=hi,c^2=gh,d^2=h,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, 7627729);