#472390

order	512
nilpotency class	3
isoclinism	isoclinic class of #455110
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1089 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

2^9|b^2=egh,c^2=f,d^2=ei,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=di,d^c=de,g^c=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, 379586);