#477149

order	512
nilpotency class	3
isoclinism	isoclinic class of #476092
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

846 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 384345);