#7891232

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	#40604	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4471 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)(9,10,14,17)(11,12,19,15)(13,16,22,24)(18,20,23,21)(25,26,30,34)(27,33,38,31)(28,32,39,29)(35,40,37,36);(1,3,5,8)(2,6,4,7)(9,10)(11,12)(13,16)(14,17)(15,19)(18,20)(21,23)(22,24)(25,27)(26,31)(28,37)(29,36)(30,38)(32,40)(33,34)(35,39);(9,11,14,19)(10,15,17,12)(13,18,22,23)(16,21,24,20)(25,28)(26,32)(27,35)(29,34)(30,39)(31,36)(33,40)(37,38);(9,11)(10,12)(13,18)(14,19)(15,17)(16,20)(21,24)(22,23)(25,29,30,32)(26,28,34,39)(27,36,38,40)(31,37,33,35);(1,2,5,4)(3,6,8,7)(9,12)(10,11)(13,21)(14,15)(16,23)(17,19)(18,24)(20,22)(25,28)(26,32)(27,37)(29,34)(30,39)(31,40)(33,36)(35,38);(1,4,5,2)(3,7,8,6)(9,13,14,22)(10,16,17,24)(11,18,19,23)(12,20,15,21)(25,27)(26,33)(28,35)(29,36)(30,38)(31,34)(32,40)(37,39)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=h,d^2=hi,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, 7798428);