#7977466

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	#2407	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4297 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=ghi,d^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,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, 7884662);