#394031

order	512
nilpotency class	3
isoclinism	isoclinic class of #394026
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#575	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3980 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)(3,4)(5,6)(7,8)(9,10,15,20)(11,21,25,32)(12,26,30,35)(13,18,14,19)(16,23,33,24)(17,28,36,29)(22,37,39,31)(27,34,38,40);(2,4)(9,11)(10,16)(12,27)(13,23)(14,24)(15,25)(17,34)(18,21)(19,32)(20,33)(22,28)(26,37)(29,39)(30,38)(31,35)(36,40);(9,12)(10,17)(11,22)(13,28)(14,29)(15,30)(16,31)(18,26)(19,35)(20,36)(21,34)(23,38)(24,27)(25,39)(32,40)(33,37);(6,8)(9,13,15,14)(10,18,20,19)(11,23,25,24)(12,28,30,29)(16,21,33,32)(17,26,36,35)(22,38,39,27)(31,34,37,40)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 301227);