#8075546

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	#2535	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4219 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)(9,10,15,14)(11,16,12,17)(13,18,22,21)(19,23,20,24)(26,32)(29,35)(33,36)(37,39);(1,2)(3,7)(4,8)(5,6)(9,11,15,12)(10,16,14,17)(13,19,22,20)(18,23,21,24)(25,26)(27,36)(28,33)(29,34)(30,35)(31,32)(37,40)(38,39);(10,14)(16,17)(18,21)(23,24)(25,27,31,28)(26,33,32,36)(29,39,35,37)(30,38,34,40);(1,3,5,4)(2,7,6,8)(9,12,15,11)(10,17,14,16)(13,19,22,20)(18,23,21,24)(25,28,31,27)(26,33,32,36)(29,37,35,39)(30,38,34,40);(1,3,5,4)(2,7,6,8)(9,13)(10,18)(11,19)(12,20)(14,21)(15,22)(16,23)(17,24)(25,29,31,35)(26,34,32,30)(27,37,28,39)(33,38,36,40);(1,4,5,3)(2,8,6,7)(9,14,15,10)(11,17,12,16)(13,21,22,18)(19,24,20,23)(25,30,31,34)(26,35,32,29)(27,38,28,40)(33,37,36,39)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=i,d^2=gh,e^2=g,f^2=gh,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, 7982742);