#8057862

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	#38997	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4311 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)(9,10)(11,12)(13,21)(14,15)(16,22)(17,19)(18,24)(20,23)(26,32)(28,34)(33,35)(36,38);(1,2,5,6)(3,7,4,8)(10,15)(12,19)(16,21)(20,24)(25,26,31,32)(27,35,29,33)(28,30,34,40)(36,39,38,37);(1,3,5,4)(2,7,6,8)(10,15)(11,17)(16,21)(18,23)(25,27,31,29)(26,33,32,35)(28,36,34,38)(30,37,40,39);(1,3,5,4)(2,7,6,8)(9,11)(10,12)(13,18)(14,17)(15,19)(16,20)(21,24)(22,23)(25,28,31,34)(26,30,32,40)(27,36,29,38)(33,37,35,39);(9,12)(10,11)(13,20)(14,19)(15,17)(16,18)(21,23)(22,24)(25,29,31,27)(26,33,32,35)(28,38,34,36)(30,37,40,39);(1,4,5,3)(2,8,6,7)(9,13)(10,16)(11,18)(12,20)(14,22)(15,21)(17,23)(19,24)(25,30)(26,34)(27,37)(28,32)(29,39)(31,40)(33,38)(35,36)

Polycyclic presentation:

2^9|b^2=gh,c^2=gh,d^2=gh,e^2=i,f^2=ghi,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, 7965058);