#441636

order	512
nilpotency class	3
isoclinism	isoclinic class of #419895
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

911 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=efg,c^2=ef,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,h^c=hi,e^d=ei,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, 348832);