#7902076

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	#2714	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4186 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)(11,18)(12,17)(13,14)(15,19)(16,22)(20,24)(21,23)(26,32)(27,29)(33,39)(35,36);(1,2)(3,7)(4,8)(5,6)(9,11)(10,16)(12,23)(13,21)(14,20)(15,22)(17,24)(18,19)(25,26)(27,30)(28,33)(29,34)(31,32)(35,37)(36,40)(38,39);(1,3,5,4)(2,7,6,8)(9,12,15,14)(10,17,19,13)(11,20,22,23)(16,21,18,24)(25,27,31,29)(26,30,32,34)(28,36,38,35)(33,40,39,37);(9,13)(10,12)(11,21)(14,19)(15,17)(16,23)(18,20)(22,24)(25,28,31,38)(26,33,32,39)(27,35,29,36)(30,37,34,40);(9,14,15,12)(10,13,19,17)(11,20,22,23)(16,21,18,24)(25,29,31,27)(26,34,32,30)(28,36,38,35)(33,40,39,37);(1,4,5,3)(2,8,6,7)(9,11)(10,18)(12,20)(13,21)(14,23)(15,22)(16,19)(17,24)(25,30,31,34)(26,27,32,29)(28,37,38,40)(33,35,39,36)

Polycyclic presentation:

2^9|c^2=g,d^2=h,e^2=hi,f^2=gi,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, 7809272);