#7870471

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	#44433	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4528 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)(9,10,15,17)(11,19,20,16)(12,13,22,23)(14,18,21,24)(25,26,31,28)(27,35,37,32)(29,38,40,33)(30,34,36,39);(1,2,5,6)(3,7,4,8)(9,11,15,20)(10,16,17,19)(12,21,22,14)(13,18,23,24)(25,27,31,37)(26,32,28,35)(29,36,40,30)(33,39,38,34);(1,3,5,4)(2,7,6,8)(10,17)(11,20)(12,22)(18,24)(26,28)(32,35)(33,38)(34,39);(1,3,5,4)(2,7,6,8)(9,12)(10,13)(11,21)(14,20)(15,22)(16,18)(17,23)(19,24)(25,28,31,26)(27,35,37,32)(29,38,40,33)(30,39,36,34);(1,4,5,3)(2,8,6,7)(9,13)(10,12)(11,18)(14,19)(15,23)(16,21)(17,22)(20,24)(25,29)(26,33)(27,36)(28,38)(30,37)(31,40)(32,39)(34,35);(9,14,15,21)(10,18,17,24)(11,12,20,22)(13,16,23,19)(25,30,31,36)(26,34,28,39)(27,29,37,40)(32,33,35,38)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gh,d^2=gi,e^2=gh,f^2=h,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, 7777667);