#8505994

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	#37356	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4302 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=h,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8413190);