#8435091

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	#43002	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4571 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=gh,d^2=h,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, 8342287);