#472013

order	512
nilpotency class	3
isoclinism	isoclinic class of #457096
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1338 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:

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

Polycyclic presentation:

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

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, 379209);