#394580

order	512
nilpotency class	3
isoclinism	isoclinic class of #394335
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2196 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)(9,10,15,19)(11,20,23,32)(12,24,28,17)(13,18,14,29)(16,30,33,22)(21,37,25,36)(26,39,27,35)(31,38,34,40);(9,11)(10,16)(12,25)(13,22)(14,30)(15,23)(17,34)(18,32)(19,33)(20,29)(21,28)(24,31)(26,40)(27,38)(35,36)(37,39);(2,4)(9,12,14,27,15,28,13,26)(10,17,18,35,19,24,29,39)(11,21,22,38,23,25,30,40)(16,31,20,36,33,34,32,37);(6,8)(9,13,15,14)(10,18,19,29)(11,22,23,30)(12,26,28,27)(16,32,33,20)(17,35,24,39)(21,38,25,40)(31,37,34,36)

Polycyclic presentation:

2^9|a^2=i,c^2=e,d^2=i,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,e^b=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, 301776);