#7902218

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	#2408	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4422 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)(11,17)(12,18)(13,16)(14,15)(19,21)(20,24)(22,23)(26,32)(28,35)(33,36)(37,39);(1,2)(3,6)(4,5)(7,8)(9,10)(11,12)(13,16)(14,15)(17,18)(19,20)(21,24)(22,23)(25,26)(27,36)(28,34)(29,33)(30,35)(31,32)(37,38)(39,40);(1,3,4,7)(2,6,5,8)(10,15)(11,18)(16,23)(19,24)(25,27,31,29)(26,33,32,36)(28,37,35,39)(30,38,34,40);(9,11)(10,12)(13,19)(14,18)(15,17)(16,20)(21,23)(22,24)(25,28,31,35)(26,34,32,30)(27,37,29,39)(33,40,36,38);(9,12)(10,11)(13,21)(14,17)(15,18)(16,24)(19,23)(20,22)(25,29,31,27)(26,33,32,36)(28,39,35,37)(30,38,34,40);(9,13,14,22)(10,16,15,23)(11,19,18,24)(12,20,17,21)(25,30,31,34)(26,35,32,28)(27,38,29,40)(33,39,36,37)

Polycyclic presentation:

2^9|c^2=g,d^2=hi,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, 7809414);