#7904706

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	#46080	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4254 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=h,d^2=gh,e^2=gi,f^2=hi,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, 7811902);