#8057842

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	#44417	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4252 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=h,d^2=gh,e^2=gi,f^2=hi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7965038);