#7929470

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
prime radical	#2388	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4229 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)(11,16)(12,15)(13,18)(14,17)(19,22)(20,23)(21,24)(26,30)(28,36)(31,33)(34,40);(1,2)(3,6)(4,5)(7,8)(9,11,15,18)(10,16,12,13)(14,19,23,24)(17,22,20,21);(10,12)(11,18)(17,20)(19,24)(25,26)(27,33)(28,32)(29,30)(31,35)(34,39)(36,37)(38,40);(1,3,4,7)(2,6,5,8)(9,12,15,10)(11,13,18,16)(14,20,23,17)(19,21,24,22)(25,27,29,35)(26,31,30,33)(28,34,36,40)(32,38,37,39);(9,13)(10,11)(12,18)(14,22)(15,16)(17,24)(19,20)(21,23)(25,28,29,36)(26,32,30,37)(27,34,35,40)(31,38,33,39);(9,14,15,23)(10,17,12,20)(11,19,18,24)(13,21,16,22)

Polycyclic presentation:

2^9|b^2=i,d^2=gh,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, 7836666);