#8067155

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	#51072	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4456 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:

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

Polycyclic presentation:

2^9|c^2=g,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=di,c^b=ci,e^c=ei,f^c=fh,e^d=ehi,f^d=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, 7974351);