#8075291

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

4263 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)(9,10,15,17)(11,16,21,14)(12,18,20,24)(13,19,23,22)(25,26,31,34)(27,29,28,32)(30,33,38,40)(35,37,36,39);(1,2,5,6)(3,7,4,8)(9,11,15,21)(10,16,17,14)(12,13,20,23)(18,19,24,22)(25,26)(27,29)(28,32)(30,33)(31,34)(35,37)(36,39)(38,40);(10,17)(14,16)(18,24)(19,22)(25,27,31,28)(26,32,34,29)(30,36,38,35)(33,37,40,39);(1,3,5,4)(2,7,6,8)(9,12)(10,18)(11,13)(14,22)(15,20)(16,19)(17,24)(21,23)(25,28,31,27)(26,32,34,29)(30,35,38,36)(33,37,40,39);(1,3,5,4)(2,7,6,8)(9,13)(10,19)(11,20)(12,21)(14,18)(15,23)(16,24)(17,22)(25,29,31,32)(26,27,34,28)(30,37,38,39)(33,35,40,36);(1,4,5,3)(2,8,6,7)(9,14)(10,11)(12,22)(13,18)(15,16)(17,21)(19,20)(23,24)(25,30,31,38)(26,33,34,40)(27,35,28,36)(29,37,32,39)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=i,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,e^a=ei,c^b=ci,e^c=ei,f^c=fh,e^d=ehi,f^d=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7982487);