#444004

order	512
nilpotency class	3
isoclinism	isoclinic class of #419810
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

924 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=ef,c^2=g,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,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, 351200);