#8455679

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	#45647	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4435 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:

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

Polycyclic presentation:

2^9|b^2=ghi,c^2=ghi,d^2=gh,e^2=ghi,f^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, 8362875);