#7941761

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

4450 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7848957);