#7954928

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	#40603	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4310 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)(11,16)(12,13)(14,19)(15,17)(18,24)(20,21)(22,23)(25,26,30,32)(27,29,34,31)(28,36,38,33)(35,37,40,39);(1,3,5,8)(2,6,4,7)(9,11,15,14)(10,16,17,19)(12,20,18,22)(13,21,24,23)(25,27,30,34)(26,31,32,29)(28,35,38,40)(33,37,36,39);(10,17)(12,18)(16,19)(20,22)(26,32)(27,34)(33,36)(35,40);(9,12,15,18)(10,13,17,24)(11,20,14,22)(16,21,19,23)(25,26)(27,31)(28,33)(29,34)(30,32)(35,37)(36,38)(39,40);(1,2,5,4)(3,6,8,7)(9,13)(10,18)(11,21)(12,17)(14,23)(15,24)(16,22)(19,20)(25,28)(26,33)(27,35)(29,39)(30,38)(31,37)(32,36)(34,40);(1,4,5,2)(3,7,8,6)(9,14,15,11)(10,19,17,16)(12,22,18,20)(13,23,24,21)(25,29,30,31)(26,34,32,27)(28,37,38,39)(33,35,36,40)

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,d^2=hi,e^2=gi,f^2=ghi,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, 7862124);