#7905713

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	#47565	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4343 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)(9,10,14,11)(12,17,20,15)(13,16,21,18)(19,23,24,22)(25,26,31,33)(27,30,36,32)(28,39,38,34)(29,40,37,35);(1,2)(3,7)(4,8)(5,6)(25,27)(26,32)(28,37)(29,38)(30,33)(31,36)(34,35)(39,40);(1,3,5,4)(2,7,6,8)(10,11)(15,17)(16,18)(22,23)(26,33)(27,36)(28,38)(35,40);(1,4,5,3)(2,8,6,7)(9,11,14,10)(12,17,20,15)(13,18,21,16)(19,23,24,22)(25,28,31,38)(26,34,33,39)(27,37,36,29)(30,40,32,35);(9,12)(10,15)(11,17)(13,19)(14,20)(16,22)(18,23)(21,24)(25,29,31,37)(26,35,33,40)(27,38,36,28)(30,39,32,34);(9,13,14,21)(10,16,11,18)(12,19,20,24)(15,22,17,23)(25,30)(26,36)(27,33)(28,40)(29,34)(31,32)(35,38)(37,39)

Polycyclic presentation:

2^9|a^2=h,c^2=gi,d^2=ghi,e^2=i,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, 7812909);