#465401

order	512
nilpotency class	3
isoclinism	isoclinic class of #456514
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	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

880 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 372597);