#7888965

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	#39426	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4389 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)(11,20)(12,13)(14,19)(15,17)(16,21)(18,23)(22,24)(25,26,31,28)(27,32,30,33)(29,36,38,34)(35,40,37,39);(1,2,5,6)(3,7,4,8)(9,11)(10,16)(12,22)(13,19)(14,18)(15,21)(17,20)(23,24)(25,27,31,30)(26,32,28,33)(29,35,38,37)(34,39,36,40);(1,3,5,4)(2,7,6,8)(10,17)(11,21)(12,18)(19,24)(25,27,31,30)(26,33,28,32)(29,35,38,37)(34,40,36,39);(9,12,15,18)(10,13,17,23)(11,22,21,14)(16,19,20,24)(25,28,31,26)(27,33,30,32)(29,36,38,34)(35,40,37,39);(1,3,5,4)(2,7,6,8)(9,13)(10,18)(11,19)(12,17)(14,16)(15,23)(20,22)(21,24)(25,29)(26,34)(27,35)(28,36)(30,37)(31,38)(32,39)(33,40);(1,4,5,3)(2,8,6,7)(9,14,15,22)(10,19,17,24)(11,18,21,12)(13,20,23,16)(25,30,31,27)(26,33,28,32)(29,37,38,35)(34,40,36,39)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=gh,d^2=h,e^2=gi,f^2=ghi,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, 7796161);