#7899496

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	#51524	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4387 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=gi,e^2=gh,f^2=g,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, 7806692);