#8069683

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	#46353	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4463 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)(11,17)(12,18)(13,19)(14,16)(15,22)(20,21)(23,24)(26,28)(31,35)(32,36)(39,40);(1,2,5,6)(3,7,4,8)(9,11)(10,15)(12,20)(13,21)(14,22)(16,17)(18,24)(19,23)(25,26,30,28)(27,31,34,35)(29,32,38,36)(33,39,37,40);(10,16)(13,18)(15,17)(21,24)(26,28)(31,35)(32,36)(39,40);(9,11)(10,17)(12,23)(13,21)(14,22)(15,16)(18,24)(19,20)(25,27)(26,31)(28,35)(29,37)(30,34)(32,40)(33,38)(36,39);(1,3,5,4)(2,7,6,8)(9,12,14,19)(10,18,16,13)(11,20,22,23)(15,24,17,21)(25,28,30,26)(27,31,34,35)(29,36,38,32)(33,39,37,40);(1,4,5,3)(2,8,6,7)(9,13,14,18)(10,19,16,12)(11,21,22,24)(15,23,17,20)(25,29,30,38)(26,32,28,36)(27,33,34,37)(31,39,35,40)

Polycyclic presentation:

2^9|b^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,e^a=ei,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, 7976879);