#466983

order	512
nilpotency class	3
isoclinism	isoclinic class of #454426
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

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

988 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 374179);