#8037022

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	C₂×#560	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4259 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=h,d^2=g,e^2=g,f^2=g,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, 7944218);