#7857887

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	#39813	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4396 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,d^2=gh,e^2=h,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, 7765083);