#8505362

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	#42463	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4250 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)(3,8)(4,6)(5,7)(9,10,15,12)(11,16,21,19)(13,22,24,17)(14,18,20,23)(25,26,29,30)(27,33,34,31)(28,36,37,32)(35,38,40,39);(1,3,6,5)(2,7,4,8)(9,11)(10,16)(12,19)(13,14)(15,21)(17,23)(18,22)(20,24)(26,30)(27,34)(32,36)(35,40);(9,10,15,12)(11,19,21,16)(13,17,24,22)(14,18,20,23)(25,26)(27,31)(28,32)(29,30)(33,34)(35,38)(36,37)(39,40);(1,2,6,4)(3,7,5,8)(9,12,15,10)(11,19,21,16)(13,22,24,17)(14,18,20,23)(25,27,29,34)(26,31,30,33)(28,35,37,40)(32,38,36,39);(1,4,6,2)(3,7,5,8)(9,13)(10,17)(11,20)(12,22)(14,21)(15,24)(16,18)(19,23)(25,28)(26,32)(27,35)(29,37)(30,36)(31,38)(33,39)(34,40);(1,5,6,3)(2,7,4,8)(9,14,15,20)(10,18,12,23)(11,13,21,24)(16,22,19,17)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=hi,d^2=ghi,e^2=gh,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, 8412558);