#7714576

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

4519 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7621772);