#422365

order	512
nilpotency class	3
isoclinism	isoclinic class of #417784
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

963 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 329561);