#448371

order	512
nilpotency class	3
isoclinism	isoclinic class of #419445
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	C₂³×C₄	C₂⁴
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1199 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)(12,24)(13,27)(14,19)(16,25)(17,26)(18,30)(20,32)(21,38)(22,36)(23,34)(28,33)(29,35)(31,37)(39,40);(1,2,5,6)(3,7,4,8)(9,11)(10,15)(12,21)(13,28)(14,23)(16,31)(17,32)(18,36)(19,34)(20,26)(22,30)(24,38)(25,37)(27,33)(29,39)(35,40);(1,3,5,4)(2,7,6,8)(9,12)(10,16)(11,20)(13,29)(14,26)(15,31)(17,18)(19,35)(21,23)(22,39)(24,27)(25,30)(28,37)(32,36)(33,38)(34,40);(1,4,5,3)(2,7,6,8)(9,12)(10,17,19,24)(11,21)(13,29)(14,26)(15,32,34,38)(16,18,35,27)(20,23)(22,37)(25,30)(28,39)(31,36,40,33)

Polycyclic presentation:

2^9|b^2=e,c^2=e,d^2=egi,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=de,f^b=fi,f^c=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, 355567);