#7857884

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	#38079	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4423 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)(9,10,15,17)(11,16,20,23)(12,14,19,24)(13,22,21,18)(25,26)(27,32)(28,33)(29,34)(30,31)(35,37)(36,38)(39,40);(1,3,5,8)(2,6,4,7)(9,11)(10,16)(12,21)(13,19)(14,18)(15,20)(17,23)(22,24)(26,31)(27,35)(33,36)(34,40);(1,4,5,2)(3,7,8,6)(10,17)(11,20)(12,19)(18,22)(25,27,30,35)(26,32,31,37)(28,34,38,40)(29,36,39,33);(9,12,15,19)(10,14,17,24)(11,21,20,13)(16,18,23,22);(9,13,15,21)(10,18,17,22)(11,19,20,12)(14,23,24,16)(25,28)(26,33)(27,34)(29,32)(30,38)(31,36)(35,40)(37,39);(9,14)(10,19)(11,18)(12,17)(13,16)(15,24)(20,22)(21,23)(25,29)(26,34)(27,36)(28,37)(30,39)(31,40)(32,38)(33,35)

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=g,d^2=h,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, 7765080);