#7877181

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

4372 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=h,d^2=hi,e^2=h,f^2=g,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, 7784377);