#413535

order	512
nilpotency class	3
isoclinism	isoclinic class of #395242
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1050 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)(5,6,9,11)(7,12,8,10)(13,14,18,24,20,26,19,25)(15,27,30,41,32,40,31,21)(16,33,35,22,29,39,36,42)(17,37,28,44,34,43,38,23);(2,4)(5,7,9,8)(6,10,11,12)(13,15)(14,21)(16,34)(17,29)(18,31)(19,30)(20,32)(22,37)(23,39)(24,40)(25,27)(26,41)(28,35)(33,44)(36,38)(42,43);(2,4)(5,8,9,7)(6,10,11,12)(13,16)(14,22)(15,28)(17,30)(18,36)(19,35)(20,29)(21,23)(24,33)(25,39)(26,42)(27,43)(31,34)(32,38)(37,40)(41,44);(13,17,19,38,20,34,18,28)(14,23,24,37,26,44,25,43)(15,29,30,36,32,16,31,35)(21,39,40,22,41,33,27,42)

Polycyclic presentation:

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

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, 320731);