#7870453

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	#46448	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4361 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)(9,10,15,12)(11,19,22,16)(13,23,21,17)(14,18,20,24)(25,26,32,37)(27,31,36,40)(28,35,30,33)(29,34,38,39);(1,2)(3,7)(4,8)(5,6)(9,11)(10,16)(12,19)(13,20)(14,21)(15,22)(17,24)(18,23)(25,27)(26,31)(28,29)(30,38)(32,36)(33,39)(34,35)(37,40);(1,3,5,4)(2,7,6,8)(10,12)(16,19)(17,23)(18,24)(25,28,32,30)(26,33,37,35)(27,38,36,29)(31,34,40,39);(1,4,5,3)(2,8,6,7)(9,12,15,10)(11,19,22,16)(13,23,21,17)(14,24,20,18)(25,29,32,38)(26,34,37,39)(27,28,36,30)(31,35,40,33);(9,13,15,21)(10,17,12,23)(11,20,22,14)(16,24,19,18)(25,30,32,28)(26,35,37,33)(27,38,36,29)(31,34,40,39);(9,14,15,20)(10,18,12,24)(11,21,22,13)(16,23,19,17)(25,31,32,40)(26,36,37,27)(28,34,30,39)(29,35,38,33)

Polycyclic presentation:

2^9|a^2=h,c^2=gh,d^2=gi,e^2=h,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, 7777649);