#398877

order	512
nilpotency class	3
isoclinism	isoclinic class of #395240
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1015 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 44 points:

(1,2)(3,4)(5,6,8,10)(7,11,12,9)(13,14,18,24)(15,25,28,21)(16,29,31,43)(17,33,19,23)(20,34,36,27)(22,32,38,30)(26,40,42,35)(37,41,39,44);(13,15,17,27,18,28,19,34)(14,20,23,25,24,36,33,21)(16,26,30,41,31,42,32,44)(22,35,29,39,38,40,43,37);(5,7,8,12)(6,9,10,11)(13,15,18,28)(14,21,24,25)(16,26,31,42)(17,27,19,34)(20,33,36,23)(22,37,38,39)(29,35,43,40)(30,41,32,44);(2,4)(13,16,17,30,18,31,19,32)(14,22,23,29,24,38,33,43)(15,26,27,41,28,42,34,44)(20,35,25,39,36,40,21,37)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 306073);