#7883122

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4341 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=hi,d^2=gh,e^2=h,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, 7790318);