#401870

order	512
nilpotency class	3
isoclinism	isoclinic class of #395181
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1234 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)(3,5,7,8)(10,15)(12,23)(13,25)(16,26)(17,32)(18,28)(19,33)(20,21)(24,36)(30,31)(34,35)(37,40);(1,3)(2,5)(4,7)(6,8)(9,10,13,15,14,18,25,28)(11,16,20,26,22,31,21,30)(12,17,23,27,24,32,36,38)(19,29,34,37,35,39,33,40);(3,7)(5,8)(9,11)(10,16)(12,19)(13,20)(14,22)(15,26)(17,29)(18,31)(21,25)(23,34)(24,35)(27,37)(28,30)(32,39)(33,36)(38,40);(9,12,13,23,14,24,25,36)(10,17,15,27,18,32,28,38)(11,19,20,34,22,35,21,33)(16,29,26,37,31,39,30,40)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 309066);