#7883693

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

4371 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)(9,10,15,12)(11,14,17,18)(13,20,22,16)(19,21,24,23)(25,26)(27,31)(28,32)(29,30)(33,39)(34,37)(35,40)(36,38);(1,2,5,6)(3,7,4,8)(9,11)(10,14)(12,18)(13,19)(15,17)(16,23)(20,21)(22,24)(26,30)(31,37)(32,38)(39,40);(1,3,5,4)(2,7,6,8)(10,12)(11,17)(16,20)(19,24)(25,26)(27,31)(28,32)(29,30)(33,39)(34,37)(35,40)(36,38);(9,12,15,10)(11,18,17,14)(13,20,22,16)(19,21,24,23);(1,4,5,3)(2,8,6,7)(9,13)(10,16)(11,19)(12,20)(14,23)(15,22)(17,24)(18,21)(25,27,29,34)(26,31,30,37)(28,35,36,33)(32,40,38,39);(9,14,15,18)(10,17,12,11)(13,21,22,23)(16,19,20,24)(25,28,29,36)(26,32,30,38)(27,33,34,35)(31,39,37,40)

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=ghi,d^2=h,e^2=g,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,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, 7790889);