#8075512

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	#49854	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4431 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)(11,16)(12,17)(13,18)(14,15)(19,24)(20,21)(22,23)(25,26)(27,36)(28,30)(29,34)(31,33)(32,35)(37,38)(39,40);(1,2,5,6)(3,7,4,8)(25,27,32,31)(26,33,35,36)(28,39,29,37)(30,38,34,40);(10,15)(13,16)(17,23)(21,24)(25,28,32,29)(26,30,35,34)(27,37,31,39)(33,40,36,38);(1,3,5,4)(2,7,6,8)(9,11,14,18)(10,16,15,13)(12,20,22,19)(17,21,23,24)(25,29,32,28)(26,34,35,30)(27,37,31,39)(33,40,36,38);(1,3,5,4)(2,7,6,8)(9,12)(10,17)(11,19)(13,21)(14,22)(15,23)(16,24)(18,20)(25,30,32,34)(26,29,35,28)(27,38,31,40)(33,37,36,39);(1,4,5,3)(2,8,6,7)(9,13,14,16)(10,18,15,11)(12,21,22,24)(17,20,23,19)(25,31,32,27)(26,33,35,36)(28,39,29,37)(30,40,34,38)

Polycyclic presentation:

2^9|b^2=g,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, 7982708);