#448328

order	512
nilpotency class	3
isoclinism	isoclinic class of #419466
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂×#534	C₂²
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1356 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|b^2=e,c^2=e,d^2=egi,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,f^b=fi,h^c=hi,f^d=fi,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, 355524);