#7893618

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	#47508	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4254 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gh,d^2=gi,e^2=gi,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, 7800814);