#8436887

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4214 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)(3,7)(4,6)(5,8)(9,10,14,18)(11,13,17,15)(12,16,21,24)(19,20,23,22)(25,26,31,36)(27,30,35,32)(28,29,34,33)(37,39,40,38);(1,3)(2,5)(4,8)(6,7)(9,11,14,17)(10,13,18,15)(12,19,21,23)(16,20,24,22)(25,27)(26,32)(28,37)(29,38)(30,36)(31,35)(33,39)(34,40);(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,24)(21,23)(25,28,31,34)(26,33,36,29)(27,37,35,40)(30,38,32,39);(2,6)(5,7)(9,11)(10,13)(12,19)(14,17)(15,18)(16,20)(21,23)(22,24)(25,26)(27,30)(28,33)(29,34)(31,36)(32,35)(37,38)(39,40);(1,3)(2,5)(4,8)(6,7)(9,12,14,21)(10,16,18,24)(11,19,17,23)(13,20,15,22)(25,29,31,33)(26,34,36,28)(27,38,35,39)(30,37,32,40);(1,2)(3,5)(4,6)(7,8)(9,13)(10,17)(11,18)(12,20)(14,15)(16,23)(19,24)(21,22)(25,30)(26,35)(27,36)(28,38)(29,37)(31,32)(33,40)(34,39)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=hi,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8344083);