#413376

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	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

902 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)(5,6,8,10)(7,11,12,9)(13,14)(15,27)(16,33)(17,36)(18,25)(19,24)(20,26)(21,30)(22,34)(23,37)(28,39)(29,40)(31,42)(32,41)(35,43)(38,44);(1,2,4,3)(5,7,8,12)(6,9,10,11)(13,15,18,30,20,32,19,31)(14,21,24,27,26,42,25,41)(16,17,29,28,35,38,34,37)(22,36,43,23,40,44,33,39);(1,3,4,2)(13,16,19,34,20,35,18,29)(14,22,25,33,26,40,24,43)(15,28,31,17,32,37,30,38)(21,39,41,44,42,23,27,36);(13,17,19,37,20,38,18,28)(14,23,25,36,26,39,24,44)(15,29,31,16,32,34,30,35)(21,40,41,43,42,22,27,33)

Polycyclic presentation:

2^9|a^2=ef,b^2=e,c^2=f,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, 320572);