#413330

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

1208 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=ef,d^2=h,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,g^b=gi,d^c=di,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, 320526);