#412863

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1048 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:

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

Polycyclic presentation:

2^9|b^2=f,c^2=e,d^2=h,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,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, 320059);