#8057901

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4177 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=gh,d^2=h,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, 7965097);