#7927706

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, prime radical, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3941 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=h,d^2=h,e^2=hi,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, 7834902);