#418733

order	512
nilpotency class	3
isoclinism	isoclinic class of #417601
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1193 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,g^b=gi,f^c=fi,g^c=gi,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 325929);