#7883697

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×#127	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4262 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)(27,37)(28,29)(30,34)(31,35)(32,36)(33,40)(38,39);(1,2,5,6)(3,7,4,8)(9,11)(10,14)(12,18)(13,19)(15,17)(16,23)(20,21)(22,24)(25,27)(26,33)(28,35)(29,39)(30,38)(31,34)(32,40)(36,37);(10,12)(11,17)(16,20)(19,24)(25,28,32,30)(26,29,36,34)(27,38,40,35)(31,37,39,33);(1,3,5,4)(2,7,6,8)(9,12,15,10)(11,18,17,14)(13,20,22,16)(19,21,24,23)(25,29,32,34)(26,28,36,30)(27,39,40,31)(33,35,37,38);(1,4,5,3)(2,8,6,7)(9,13)(10,16)(11,19)(12,20)(14,23)(15,22)(17,24)(18,21)(25,30,32,28)(26,34,36,29)(27,38,40,35)(31,37,39,33);(9,14,15,18)(10,17,12,11)(13,21,22,23)(16,19,20,24)(25,31,32,39)(26,35,36,38)(27,34,40,29)(28,37,30,33)

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=hi,d^2=gh,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, 7790893);