#7712402

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4388 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)(11,12)(13,21)(14,15)(16,22)(17,20)(18,24)(19,23)(25,26,32,36)(27,28,30,33)(29,34,37,40)(31,38,39,35);(1,2,4,5)(3,6,7,8)(10,15)(11,17)(16,21)(18,23)(25,27,32,30)(26,33,36,28)(29,31,37,39)(34,35,40,38);(1,3,4,7)(2,6,5,8)(9,11,14,17)(10,12,15,20)(13,18,22,23)(16,19,21,24)(25,28,32,33)(26,27,36,30)(29,35,37,38)(31,34,39,40);(25,29)(26,34)(27,31)(28,38)(30,39)(32,37)(33,35)(36,40);(9,12)(10,11)(13,19)(14,20)(15,17)(16,18)(21,23)(22,24)(25,30,32,27)(26,28,36,33)(29,39,37,31)(34,38,40,35);(9,13,14,22)(10,16,15,21)(11,18,17,23)(12,19,20,24)(25,31,32,39)(26,35,36,38)(27,37,30,29)(28,34,33,40)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7619598);