#7927776

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

4617 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=h,d^2=i,e^2=gi,f^2=i,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, 7834972);