#7734368

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	#2847	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4510 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=g,e^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7641564);